For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$. These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$. In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas. Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.