cycle structure of bounded genus graphs

Question

Given fixed genus $k$ is there some $q=f(k)$ for which for any graph $G$ of bounded genus $k$ there are $q$ planar subgraphs $G_1,G_2,..,G_q$ of $G$ such that each cycle of $G$ can be written as the symmetric difference of cycles in $G_i$?
If true what is a bound of $f(k)$ can you use a fixed (non-planar) embedding of $G$ such that each $G_i$ is planar? If not is there a condition that can be relaxed?