Can I fix my husband's papers if he gave the wrong name to immigration?

What do you mean he gave the wrong name? do you mean he entered the US with fraudulent documents?

As a US citizen, you can petition for your husband But prior to taking such a step, both you and your spouse must consult with an experienced immigration lawyer. You stated that he got papers from immigration using a wrong name. You need to give a more detailed information regarding the background facts and circumstances of his previous contacts with US immigration. At a bare minimum, there is an issue of his identity which must be resolved first. Or else, what name will he use, if you file paperwork for him. Regardless of the name he now wishes to use, he must undergo biometrics processing which is likely to return a "hit" if a previous information regarding him (whatever previous aliases previously) is already on file. Although, this is a fairly complicated case, rest assured it can be resolved to you and your husband's satisfaction. However, to do so , you need to seek advice of a competent counsel.

It depends on a lot of factors, but trying to fix his papers may expose him to a lot of unwanted consequences. He may even be found inadmissible and placed into removal proceedings.