Question: What is the relationship between provability in $RCA\_0$ and effectively true?

In other words: Given a problem, if a statement asserting the existence of a solution of the problem is provable in $RCA\_0$, does it follow that given a computable instance of the problem we can compute a solution to the problem?

Motivation: I'm getting into Reverse Mathematics and Hirchfeldt gives an exercise in his book Slicing the Truth to prove that $RCA\_0$ proves the Ramsey theory for singletons $RT^1\_k$ for each $k \geq 2$. This together with $REC$ being the intended model for $RCA\_0$ implies to me that the homogenous set should be computable.

However, $RT^1\_k$ seems to be effectively true only in the trivial case of $k=1$.
My thinking so far got me the following observations:

1. If the colour in the infinite pigeonhole principle is effectively true, then so is $RT^1\_k$, but this doesn't seem to help, since the two are equivalent.
2. Trying to define the property that the homogenous set is infinite leaves me with the solution being $\emptyset''$-computable.

To my limited knowledge the most probable solution to this, unless I'm missing some trick that would make $RT^1\_k$ effectively true, is that while homogenous sets are computable, we cannot effectively decide which homogenous set is infinite, hence cannot effectively compute the solution to the instance.

My naive leap from this is that provable in $RCA\_0$ implies effectively true only with finite and trivial instances. But this seems quite a long stretch and the use of trivial is rather vague(to incorporate the $RT^1\_k$ case, something along the lines of Rice's theorem) hence the question.

(Apologies if this is not a research-level question, my competence to evaluate these is lacking)