translation - In the next sentence she wants to heal her wounds herself or she wants them to be healed?

The sentence is as follows:

私が付けた傷が、少しでも癒えますようにって。

In hopes that the pain I've caused you will be eased, even if it's only by a tiny bit.

Or

Let me heal the wounds that I caused you, even a little.

The previous sentence was あなたのことを気遣ってると思いこむことで、逃げてるんだよ。

Answer

I'm leaning toward the former. The important thing here is the transitivity of the verb is - we have 癒える and not 癒やす, with the former being intransitive (自動詞) and the latter being transitive (他動詞). For some very simplified example sentences, you'd see usages like

私の傷が癒えます (My wounds will heal)

vs

私はあなたの傷を癒やします (I will heal your wounds)

Consequently, the way it's written, it sounds like it's about wounds healing on their own. At the very least if the author wanted to express the idea that the subject of the sentence was going to heal those wounds, they intentionally chose not to.