  A graph where each vertex $v$ has a list $L(v)$ of available colors is
$L$-colorable if there is a proper coloring such that the color of $v$ is in
$L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at
least $k$ colors to each vertex guarantees an $L$-coloring. Given a list
assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this
paper, we look at a variant of the widely studied class of precoloring
extension problems from [Z. Dvo\v{r}\'ak, S. Norin, and L. Postle: List
coloring with requests. J. Graph Theory 2019], wherein one must satisfy
"enough", as opposed to all, of the requested set of precolors. A graph $G$ is
$\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$,
and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an
$\varepsilon$-fraction of the requests in $S$. It is conjectured that planar
graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for
list size $6$ and for certain subclasses of planar graphs. We give a stronger
version of the main tool used in the proofs of the aforementioned results. By
doing so, we improve upon a result by Masa\v{r}\'ik and show that planar graphs
without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove
that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are
$\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly
weaker) form of $\varepsilon$-flexibility where each vertex has exactly one
request. In that setting, we provide a stronger tool and we demonstrate its
usefulness to further extend the class of graphs that are
$\varepsilon$-flexible for list size $5$.
