  In this paper we consider three types of discrete operators stemming from
singular Radon transforms. We first extend an $\ell^p$ result for translation
invariant discrete singular Radon transforms to a class of twisted operators
including an additional oscillatory component, via a simple method of descent
argument. Second, we note an $\ell^2$ bound for quasi-translation invariant
discrete twisted Radon transforms. Finally, we extend an existing $\ell^2$
bound for a closely related non-translation invariant discrete oscillatory
integral operator with singular kernel to an $\ell^p$ bound for all $1< p<
\infty$. This requires an intricate induction argument involving layers of
decompositions of the operator according to the Diophantine properties of the
coefficients of its polynomial phase function.
