Even if the answer is no, I am interested in a more specific question.

Let $\Sigma$ be a set of operations of finite arity, $E$ be a set of equations over $\Sigma$ and $\mathcal{A}(\Sigma,E)$ be the respective category of algebras and algebra morphisms. Also denote the free algebra functor by $F: \mathsf{Set} \to \mathcal{A}(\Sigma,E)$.

If $f : A \to B$ is a monomorphism in such a category i.e. an injective algebra morphism, and also $X$ is set, does it follow that $FX + f : FX + A \to FX + B$ is also injective?

Any help much appreciated.