" it can be used to argue that any complex dimension 2 torus cannot embed in CP4" That is not correct. There are abelian surfaces embedded in P^4, namely the zero-loci of sections of the Horrocks-Mumford bundle. It is true that a complete intersection in P^N cannot be a complex torus.

For sure one can construct examples of pairs $(X,m)$ for which the answer to your yes, but there are also examples where $Y_m$ does not contain any rational curves. What kind of answer are you looking for?

Hmm, it seems I cannot delete (or even flag for deletion?) my own answer. Oh well.

OK, in this case I think you should unaccept my answer and I will delete it for the time being.

@AriyanJavanpeykar: sure, all correct, let me fix.

@DamianRössler: that's a nice reformulation, thanks.

@DamianRössler: I claimed a bit more than my proof shows. I wrote an answer below to elaborate on what I had in mind.

If I understand the terminology correctly, no. There are examples of 2-dimensional compact complex manifolds $X$ which contain no compact analytic subvarieties of dimension 1. For such an $X$, take $A$ to be any point in $X$.