Let $U\subseteq\mathbb{C}^n$ be open and $0$-connected, and let $\Gamma\in Z_1(U,\mathbb{Z})$ be a $1$-cycle in $U$. Let $p_k:\mathbb{C}^n\to\mathbb{C}$ be the $k$-th projection, it is continuous and open and induces a map in homology $(p_k)_*:H_1(U,\mathbb{Z})\to H_1(p(U),\mathbb{Z})$.
I am looking for conditions on the projected $1$-cycles $\Gamma_k:=(p_k)_\#\Gamma$ that would ensure $[\Gamma]=0$. Clearly, if $[\Gamma]=0$, then $[\Gamma_k]=0$, so it is a necessary condition, but not sufficient, unless $U$ is so nice that $\bigcap_{1\leq k\leq n}\ker(p_k)_* = 0$.
If $U$ were a product, then one could use Künneth formula. In general, $U$ can be covered by a countable famility of products of opens, and I imagine there is a countable version of Mayer-Vietoris, but this seems like an overkill approach. On the other hand, open sets can be very complicated, so maybe it's hopeless?
If it helps, you can assume that $U$ is bounded.
EDIT: Nick L's comments made me realize that the previous title formulation was ambiguous.
I am looking to express the condition $[\Gamma]=0$ through $\Gamma_1,\dots,\Gamma_n$ rather than $[\Gamma_1],\dots,[\Gamma_n]$.
Sorry for my oversight!
The following example illustrates that the condition $[\gamma] = 0$ cannot be recovered from the $[\gamma_{k}]$. In the following example there are infinitely many non-zero $[\gamma]$ for which $[\gamma_{k}] = 0, \forall k$.
Let $U$ be the complement of set $\{z_{1}^{3} + z_{2}^{2} = 0\}$ in $\mathbb{C}^{2}$, intersected with an open ball of radius $r>0$ (centred at $(0,0)$). Then $U$ is diffeomorphic to $$ (S^{3} \setminus K) \times (0,r), $$ where $K$ is the trefoil knot*. Hence, $H_{1}(U , \mathbb{Z}) \cong \mathbb{Z}$ (this follows because the homology of any knot complement in $S^{3}$ is $\mathbb{Z}$ and the Kunneth formula).
The image of $U$ by $p_{i}$ is an open ball in $\mathbb{C}$ with radius $r$ (centred at the origin), hence $H_{1}(p_{i}(U,\mathbb{Z}))= \{0\}$, for $i$ = 1,2.
*See here https://en.wikipedia.org/wiki/Torus_knot (section: connection to complex hypersurfaces) for a description of the trefiol knot as the link of the singularity $z_{1}^{3} + z_{2}^{2} = 0 $.
Perhaps there is something that can be said if one imposed some additional assumptions on $U$... In any case, I thought this example might be helpful and it was too long for a comment.