Please, is this sentence correct? 'He has an husband already.' Or should it be 'he has a husband?'

It's "a husband."

Here's more on an or a.

he has a husband

the correct sentence is:
she has a husband.
There is one more thing:
a, an are definite articles, we cannot use it anywhere.
Ex: a used before consonant sounds " I have a book. "
an used before vowel sounds (a, i, o, u, e) " I eat an apple every day."