  We analyse a collection of twisted mixed moments of the Riemann zeta function
and establish the validity of asymptotic formulae comprising on some instances
secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$
and a polynomial $P(x)$. Such examinations are performed both unconditionally
and under the assumption of a weaker version of the $abc$-conjecture.
