  For all positive powers of primes $p\geq 5$, we prove the existence of
infinitely many linear congruences between the exponents of twisted Borcherds
products arising from a suitable scalar-valued weight $1/2$ weakly holomorphic
modular form or a suitable vector-valued harmonic Maa{\ss} form. To this end,
we work with the logarithmic derivatives of these twisted Borcherds products,
and offer various numerical examples of non-trivial linear congruences between
them modulo $p=11$. In the case of positive powers of primes $p=2,3$, we obtain
similar results by multiplying the logarithmic derivative with a Hilbert class
polynomial as well as a power of the modular discriminant function. Both
results confirm a speculation by Ono.
