Usual signatures do not allow homomorphic operations on the signed values. You will have to use special fully homomorphic signatures (FHS) for that.

Suppose you did this and encrypted + signed your values with FHE and FHS as Sign(Encrypt(x)) and Sign(Encrypt(y)). Now you want to add x and y. For that, you will have to run the homomorphic addition of Encrypt(·) values using homomorphic operations of Sign(·), i.e. you do Encrypt(x) + Encrypt(y) inside Sign(·), not just x + y.

This means that the overheads introduced by FHE and FHE do not just add up, they get composed (as in composition of functions). This will be a terrible blow-up in performance. I guess, for this reason people try to build authenticated FHE that will have the properties of FHE + FHS but as a one, more efficient block.

Then your input is signed. And now what? How does this authenticate the output?