  We show that every sufficiently large integer is a sum of a prime and two
almost prime squares, and also a sum of a smooth number and two almost prime
squares. The number of such representations is of the expected order of
magnitude. We likewise treat representations of shifted primes p-1 as sums of
two almost prime squares. The methods involve a combination of analytic,
automorphic and algebraic arguments to handle representations by restricted
binary quadratic forms with a high degree of uniformity.
