Settlers of Catan
In Settlers of Catan (SoC), there are two types of cards: resources and developement cards. I will discuss the developement cards, as the resource cards are all identical (the resource cards are not variable power cards). There are 25 available cards: fourteen Soldiers that move the robber, two each of three cards that give you bonuses (roads, resource cards from the deck, resource cards from other's hands), and 5 victory point cards.
All cards are useful to some degree at any time. A VP cards may not be that useful at the beginning of the game, but will surely be by the end of the game. A soldier may is always worth at least one resource card stolen from your opponent, even if you have no need of moving the robber, and is a step in securing two bonus victory points. The other cards are each worth their cost, except in the rare case when you have no more roads to build.
The game can be played quite well without these cards. Players can build roads, settlements and cities quite well. The game involves a random element of dice rolls which has a high luck factor; you can plan for the best spots on the board. You can work around some poor rolls with good trading. You can position yourself not to need certain resources more than others. Ultimately, a long string of bad luck, or bad luck at certain times, will kill your game.
The cards add the following elements to the game: a) additional avenue of victory, through Larget Army and the VP cards, b) opportunity to plan against the robber blocking your resources, c) opportunity to slow down the leader (with some luck), d) opportunity to win when other avenues are closed to you, such as settlement space on the board, e) resources that might be too hard to acquire otherwise.
In Amun-Re, power cards are one of the three things you can buy each round, along with bricks and farmers. There is no limit to the number of bricks you can buy, a space limitation on farmers, and a purchase limitation on cards. The cards give you: free farmers, free bricks, vp's if you fulfill certain requirements, slight control over the auction or slight control over the group "sacrifice" action. Any card can be discarded for 1 GP.
Many of these cards are simply more powerful than others, such as getting a free Farmer vs "bid again in the same province". Some cards are useless in certain situations, such as duplicates at the end of the game, vp cards when you can't fulfill the requirements, etc...
The game can be played well without the cards. Players can build pyramids, and buy farmers, acquire gold and auction. The random element of the game is acheived through the uncertainty of the auction and the random order in which the regions are available.
The cards add the following elements: a) additional vp's through region matching, and acquiring some types of resources, b) some power to manipulate the auctions via a surprise element, c) free resources that could be acquired anyway. In my opinion, the detracting elements of the cards outweighs the benefit in this game. a) could be achieved by having players buy what vp cards they wanted directly, or make them available through some other means, or simply by giving all bonuses met to all players. The b) cards are not powerful enough to warrant buying them; each player should simply start with one of each to be used once during the game. If the remaining cards consisted only of c) cards - free bricks, farmers or money - the game would be better off.
Practical Solution: when you buy a card, pick two, keep one.
In Evo, power cards have numerous different effects. Three are dealt to each player and others may be acquired at the expense of adaptations through the auctions. Most cards simulate a double or triple power adaptation for the round they are played in, some cards change the climante direction in a manner that benefits your planning. A few cards are useless. A few cards are heavily luck dependent.
The game can be played well without the cards. Players can auction off the adaptations, move, fight, etc... withou regards to the cards. They seem to server no function except to provide surprise. They are used to make the game less strategic and more wild, and to simulate uncertainty.
As suggested on the Geek, the cards are better off being picked by the player, instead of being dealt randomly.
In Cosmic Encounter, there are two types of cards in the same deck. Challenge cards are the dice rolling of the game. I will not concern myself with these. The remaing cards, Edits, Flares, Kickers, etc... are the power cards. The provide a limitless set of different experiences in the game.
While the game can be played without these cards, as the powers are already a random factor, one of the joys of Cosmic is that the cards are so over the top that they far outweigh all tactics in the game. Powerful cards will certainly determine the victor. There are so many of them however, and the interact in so many non-intuitive ways, that you can never be completely certain that the card you play is going to be the one to do it.
In a sense, it is specifically the surprise element of the cards that drives the game, as the game itself is of no strategic interest, except in regards to the formation of allies and laying out of the spotlight.
Princes of Florence
In Princes Of Florence (PoF), there are three types of cards: professions, bonuses and prestige cards. Profession cards are played for indirect points; their value depends on what areas you have invested in. You can alleviate the randomness of these cards by buying them early and planning around them. Bonus cards add between 2-4 indirect points, and you choose 1 card out of 5. The difference is usually a 1 point difference, and negligible to the game, except insofar as it does or doesn't let you play a work that you had not planned for correctly. Prestige cards give points directly, and again you choose 1 out of 5. They are typically worth between 5-8 points, and rarely you will not be able to select a worthwhile card. Again, buying them early allows you to plan around them, although this is not always easy. Buying them late introduces a large element of luck.
The game cannot be played without the profession cards, and thankfully the luck issue in them is irrelevent. The game could be played without the other cards. You couls score points by playing works.
The bonus cards add an element of strategy to catch up when you have no alternatives for playing a card, and for trying to secure best work in a round. The prestige cards introduce another means of acquiring vp's.
It is likely that the game would be slightly less luckier if the prestige cards were no longer available during the last two rounds, or if one could choose which ever cards they wanted during the last two rounds.
The power cards of Goa provide more of the resources that can otherwise be provided (but may be difficult to do so), as well as colonists which cannot be readily acquired. In general, they may be planned for. Unfortunately, many of the cards are stronger than others, especially as the game progresses. Colonist cards are useless after you have acquired all of your colonies. Progression for money is nigh impossible in a cash poor game. Due to the hand limit, which card you get when is a matter of high luck. If you have good ship production, getting more ships is less useful than getting spices, and vice versa.
The solution is to buy them before any other actions, and then plan your actions accordingly. This is not always possible, and some cards may go to waste.
The power cards are also used directly for victory points my matching symbols on the cards. If you match the symbols, you score more points, otherwise less points. Since a game of Goa is often won by 1-2 points, this introduces a high luck factor into the game.
The game could be played well without the cards and the card track, by reducing the number of auctions to six, for instance. The auctions provide a high degree of randomness with little luck (but we won't go into that).
The cards add: a) the possibility of acquiring items that are beyond your reach otherwise, and quicker, b) an additional means of vp's to win the game, c) colonists that could otherwise not be acquired.
I have written a long series of changes I would like to make in this game, elsewhere. Just addressing the above, you could remove all of the cards, change the track to acquiring colonist cards, and put symbols on the colonist cards which you could choose when you buy them.
These games have many things in common. The first four games involve allocating your pieces onto the board to control certain positions. The last two games require you to collect and play various items in order to advance your position independent of other's activities. The power cards also seem to be very similar, adding free items that are otherwise hard to acquire, or additional vp's. However, in each game, the game's balance indicates how strong a role the cards play in determining whether the cards are used to add a high degree of luck or simply randomness into the play.