For a primes $p$ sufficiently large, does there always exists positive integers $k,a,b\in\mathbb{N}$ such that $p=(k+1)(ab)+k(a+b)$ or equivalently $p\equiv (ab)\bmod ((a+b)+ab)$?

Please note that from Frobeneius $p=(k+1)X+kY$ is solvable in non-negative integers $X,Y$ provided p>XY-(X+Y), what I am looking for is the existence of a sub-class of positive intger solutions where X=ab and Y=a+b.