  We prove a version of the Bombieri--Vinogradov Theorem with certain products
of Gaussian primes as moduli, making use of their special form as polynomial
expressions in several variables. Adapting Vaughan's proof of the classical
Bombieri--Vinogadov Theorem to this setting, we apply the polynomial large
sieve inequality that has been recently proved and which includes recent
progress in Vinogradov's mean value theorem due to Parsell et al. From the
benefit of these improvements, we obtain an extended range for the variables
compared to the range obtained from standard arguments only.
