Definability over $\mathrm B\Sigma^0_2$-models

Let $\mathfrak M=(M,\mathcal X)$ be a model of \mathsf{RCA}_0+\text{\Sigma^0_2-bounding} in which $\Sigma^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a $\Delta^0_1(A)$-instance of the principle with no solution in $\mathfrak M$ that is arithmetically definable relative to $A$; and (ii) any set of minimal Turing degree in $\mathfrak M$ that is arithmetically definable relative to $A$ has Turing jump equivalent to $A'$.