The Hardy-Littlewood problem asks for the number of
representations of an integer as the sum of a prime and
two squares. We consider the Hardy-Littlewood problem where
the two squares are restricted to squares of
almost primes. A lower bound of the expected order of
magnitude can be obtained. The same technique also
shows that there are infinitely many primes that can be written
as sum of two almost prime squares plus one.
We also discuss the problem of writing an integer as the
sum of a smooth number and two almost prime squares.
This is based on joint work with V. Blomer, L. Grimmelt and S. L. Rydin Myerson.