If $S_g$ is the surface $\#_g T^2$ where $g$ is a non-negative integer, when can we construct a covering space $S_h$ of $S_g$?
Each such surface is a $CW$-complex, and in a $n$-sheeted covering, each $0$-cell, $1$-cell, and $2$-cell lifts to $n$ such cells; thus if $S_h$ is an $n$-sheeted covering of $S_g$ $\chi(S_h) = n \chi(S_g)$.
Thus it is necessary that there exists $n \ge 1$ such that $h = n(g-1) + 1$. Assuming that this condition is met, it it always possible to construct such a covering space?
To elaborate on Mike Miller's comment, it is easy to see that if $p:X\to Y$ is an $n$-sheeted cover, then $p$ induces a covering $q:X\#Z\#\dots\#Z\to Y\#Z$, where there are $n$ copies of $Z$ on the left. Indeed, if you remove a ball $B$ from $Y$, then $p$ restricts to a covering map from $X$ with $n$ balls removed to $Y\setminus B$; gluing in copies of $Z$ with a ball removed everywhere then yields the covering map $q$. To get your desired covering map, you now just have to let $X=Y=T^2$ and $Z=S_{g-1}$, and observe that there exists an $n$-sheeted cover $T^2\to T^2$ for all $n$.