Lagrange's four squares theorem with one prime and three almost--prime variables

It is conjectured that every sufficiently large integer $N\equiv 4\pmod{24}$ should be a sum of the squares of 4 primes. The best approximation to this in the literature is the result of Brüdern and Fouvry [J. Reine Angew. Math., 454 (1994), 59--96] who showed that every sufficiently large integer $N\equiv 4\pmod{24}$ is a sum of the squares of 4 almost-primes, each of which has at most 34 prime factors. The present paper proves such a result with the square of one prime and 3 almost-primes, which in this case have at most 101 prime factors each. The work of Brüdern and Fouvry was based on Kloosterman's approach to representations by quaternary forms, but this does not lend itself to situations in which one of the variables is restricted to be a prime. Instead the present paper works with an `almost all' result for the representation of numbers $m$ as sums of 3 squares. To use this approach one has to take $m$ of the form $N-p^2$, and such numbers are too sparse for the standard theory. It is therefore necessary to use an `amplification' procedure, which emphasizes those integers $m$ for which $N-m$ is a square. All this machinery is coupled with Kloosterman's version of the circle method. There are considerable technical complications, in which bounds for the Kloosterman sum play a key rôle. At one point in the argument a saving has to be extracted from a non-trivial averaging over the denominators of the Farey arcs. This is an instance of `the second Kloosterman refinement'