Consider a real-valued $Y$ and a Hilbert-space valued $X$ such that $Y|X\sim N(0,1)$. Let $(Y+c_X,X)\sim P_1$ and $(Y,X)\sim P_0$, where $c_X$ is a function of $X$. Show that $P_1\ll P_0$.
My attempt: for every $A,B$ $$P_1(A\times B) = \Pr(Y+c_X\in A,X\in B) = \Pr(Y+c_X\in A|X\in B)\Pr(X\in B).$$
Then if $P_0(A\times B)=0$, we should also have $\Pr(Y+c_X\in A|X\in B)\Pr(X\in B)=0$ since $Y|X$ has non-zero probability on the entire real line.
Does this make sense?
I think you also need to be more careful.
The conditional distribution you wrote are absolutely continuous with respect to a standard gaussian because it is just a shifting of the pdf.
Since we can write $P_0$ using bayes rule (as you did for $P_1$), then we see that if it is 0, then at least one of the terms must be 0, either the conditional or the marginal. The marginals are equal in both cases, and we already showed mutual absolute continuity of the conditionals. So we are done.