Additive uniqueness of $\mathtt{P}\mathtt{R}\mathtt{I}\mathtt{M}\mathtt{E}\mathtt{S}-1$ for multiplicative functions

Let $\mathtt{P}\mathtt{R}\mathtt{I}\mathtt{M}\mathtt{E}\mathtt{S}$ be the set of all primes. We show that a multiplicative function which satisfies $$f(p+q-2)=f(p)+f(q)-f(2)\text{ for }p,q\in \mathtt{P}\mathtt{R}\mathtt{I}\mathtt{M}\mathtt{E}\mathtt{S}$$ is one of the following: \begin{enumerate} \item $f$ is the identity function \item $f$ is the constant function with $f(n)=1$ \item $f(n)=0$ for $n\ge 2$ unless $n$ is odd and squareful. \end{enumerate} As a consequence, a multiplicative function which satisfies $$f(a+b)=f(a)+f(b)\text{ for }a,b\in \mathtt{P}\mathtt{R}\mathtt{I}\mathtt{M}\mathtt{E}\mathtt{S}-1$$ is the identity function.