Universal sums of generalized polygonal numbers of almost prime length

In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in {\mathbb{N}}_{\ge 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of prime divisors (counting multiplicity) away from an finite set of exceptional primes. In the first theorem, we fix $m$ and uniformly bound the finite check independent of $L\ge 900$, and in the second theorem, we give an optimal bound for the finiteness check if $L$ is larger than a constant times $\log (m)$.