This problem is similar to Upper bound on difference of expected values between two close distributions, and we use the same notations. Let $p$ and $q$ be two distributions over the same finite set $X$ and the total variation distance $d_{TV}(p,q)$ is bounded by $\epsilon$.
Q: For some function $f(x): X \to \mathbb{R}$, is the difference of expectations $\left|\sum_x f(x)(p(x)-q(x))\right|$ can be bounded by the expectation of $p$ (or $q$), i.e., $\left|\sum_x f(x)p(x)\right|$?
In the original post, @Adam gives a bound for the difference of expectations with respect to the $\max |f(x)|$, I wonder whether there exists a tighter one by using the expectation of $p$ (or $q$).