There seem to be a few confusions implicit in your question, so let me try to explain what is going on here before addressing your explicit questions.
Let's consider what happens for a function $f$ defined as in your first expression. One proves the function $f$ is recursive by showing how to compute it, in the following way: On input $n$, begin enumerating $A$ (which can be done recursively because $A$ is recursively enumerable). If ever $n$ appears in the enumeration, then we know that $n \in A$: in that case, compute and output $g(n)$ (a recursive function). Since $n$ will eventually be enumerated if and only if $n\in A$, this means that our procedure outputs $g(n)$ exactly when $n\in A$, and our procedure never halts exactly when $n\not\in A$. Thus this procedure in fact computes $f(n)$, so $f$ is a (partial) recursive function.
Note that this construction doesn't work, and thus we can't conclude that $f$ is recursive, if we need our program to output something when $n\not\in A$. If $A$ is not recursive (but only recursively enumerable), we can never be satisfied, no matter how much of $A$ we enumerate, that some $n\not\in A$. So we won't be able to rely on an enumeration of $A$ to decide what to output depending on whether $n\in A$, because if we ever chose to guess that $n\not\in A$ (having not seen it so far) we could potentially be wrong. This is why we need $f$ to be undefined for $n\not\in A$ if we are to conclude that $f$ is recursive.
Lastly, you ask what are the conditions for $h$ under which a function $f$, which is defined to be $f(n)=g(n)$ for $n\in A$ and $f(n)=h(n)$ otherwise, is recursive. It's not really possible to give a general answer to this that doesn't just restate the problem. Certainly some sure $h$ work: for instance, $h=g$ (because then $f = g$ as well, and thus $f$ is recursive because $g$ is). In general there could be many such $h$, because there are many recursive functions which coincide with $g$ for values in $A$. Describing all such functions is essentially impossible. What you cannot do (at least if $A$ is not recursive itself) is find such a function $h$ such that $g(n)\neq h(n)$ for all (or all but finitely many) $n$. If you could, then you could determine (by computing $f(n)$, $g(n)$, and $h(n)$) whether or not $n\in A$, and hence $A$ would be recursive.