Can you use the Magic Lamp after you have used a fate. The card first calls it's action a wish and later goes on to describe it as changing the roll by using a fate. The card seems to have two meanings.

You can only use fate "once per die roll".

If you have already used a fate token of your own to re-roll a die you cannot then use a fate from the Magic Lamp to re-roll the die again.

Best just to use the Magic Lamp to begin with and therefore guarantee the outcome!

It's confusing because the fate is defined as a wish in the first sentence of the card.

Let's have a closer look:

Magic Lamp

The genie of the lamp is slave to no man for long; use your wishes wisely. Place 3 fate on the Magic Lamp when you draw it. After you roll, you may spend 1 fate from this card to change the result of 1 die to a result of your choice. When there is no more fate on this card immediately place 3 fate on the Magic Lamp and pass it to the character to your left.

First sentence is flavour text, so just ignore it if it confuses you.

The card says to use 3 fate tokens as counters to place on the lamp (of course they are not fate points that can be spent like fate). They are spent as instructed by the card, which means they change a roll to the result of your choice (not re-rolling).

You may always use the Magic Lamp to change the result of a roll. You may even do this when the roll has been rerolled with fate, or even if Misfortune has been cast and the die has automatically rolled a 1. The instructions "cannot be rerolled" are not applicable here, because you're not rerolling anything.

A small clarifications from FFG is perhaps needed to confirm whether a result rerolled with fate can be altered with the Lamp, since the rules say that " If a player pays a fate token to reroll a die, he must accept the new result; he may not pay another fate token to reroll the same die again ". I believe this rule has been added to avoid multiple rerolls of the same die, but I don't think it should limit the Magic Lamp in any way.

Your concept is the same as mine in relation to the fate tokens being used as counters and not, in spirit, fate at all.

That being established, a wish from the lamp, would certainly be able to change your fate roll to whatever number you see fit.

You can only use fate "once per die roll".

If you have already used a fate token of your own to re-roll a die you cannot then use a fate from the Magic Lamp to re-roll the die again.

Best just to use the Magic Lamp to begin with and therefore guarantee the outcome!

The fate isn't being re-rolled. The wish will change the fate roll to whatever number you wish it to be. So there is no double rolling your fate.