Let $X, Y$ be random variables, and $X$ positive and bounded ($0\leq X <C$ for some constant $C$).
Is it true that $$\lvert E(XY)\rvert\leq c\lvert E(Y)\rvert$$ for some constant $c>0$? If yes, why?
Comment: I can only obtain that $\lvert E(XY)\rvert\leq C E\lvert Y\rvert$.
Your comment contains a hint why the answer is "no".
If $X$ is a non-degenerate random variable, you can find constants $a$ and $b$ so that $Y=aX+b$ obeys $E[XY]=1$ and $E[Y]=0$. Then the only $c$ satisfying your inequality is $c=0$.