A related question asked about intermediate problems between $L$ and $NL$. I'm interested in problems between the nonuniform versions of $NL$, $Mod_n L$ and $P$. This makes sense as it is known that $NL \subseteq Mod_2 L$ nonuniformly (see here).
A motivation comes from the complexity classification of Constraint Satisfaction Problems. To the best of my knowledge, all known dichotomies in this field yield tractable cases that fall in one of the above classes. A seminal result in this area is Schaefer's dichotomy for binary CSPs showing that the tractable instances are reducible to either $2SAT$ (thus $NL$-complete), $HornSAT$ or its dual (thus $P$-complete) or satisfiability of affine equations (thus $\oplus L$-complete).
This question can also be declined in two variants: (i) what about the corresponding counting classes, (ii) what about the corresponding parameterized classes, e.g. parameterized problems non-deterministically solvable in $O(k \log n)$ space vs problems solvable in $n^{O(k)}$ time.