You are right to be confused: the sentence is not clear or grammatically correct. I believe the intention is that a horse will be gentle only if it has been well-trained. I suggest that when you answer the question you clearly state what sentence you are translating. Otherwise, the grader might mark down your answer because they have a different idea of what "Only horses are gentle if they have been well trained" means.
But note that your proposed translation of "horses are the only thing that can be made gentle by training" is incorrect: it says that whenever $T(x)\to G(x)$ is true for $x$, then $x$ is a horse. $T(x)\to G(x)$ is true if $x$ is something that is both trained and gentle, but it is also true if $x$ is anything untrained. So your translation also claims that anything that has not been trained is a horse.
Note that $T(x)\to G(x)$ does not mean "$x$ becomes gentle by training". It is equivalent to $\lnot T(x)\lor G(x)$, which says that $x$ is untrained or gentle. Your translation is equivalent to:
$$ \forall x [ (\lnot T(x) \lor G(x))\to H(x)]$$
which says that any $x$ which is untrained or gentle, must be a horse. This surely wasn't what you meant to say, since it says that untrained lemurs and gentle octopuses are horses, and there was no trace of that in the English version.
I don't think it is possible to translate "horses are the only thing that can be made gentle by training" propositionally. The problem is that you don't have any relation that expresses that the gentleness is the result of the training.
I suggest you try "the only horses that are gentle are those that have been well-trained", as I did, or "nothing well-trained is gentle, except for horses" as Peter Smith did elsewhere. My own translation of "the only horses that are gentle are those that have been well-trained" is:
$$\forall x [(H(x)\land G(x))\to T(x)]$$
$$\forall x [H(x)\to (G(x)\to T(x))]$$
(Mouse over for spoiler.)