Nsryjdtyk

I’m reading a manga and I’m a little confused to the meaning of きれる in this context.

“甘さを自覚した三浦くんは

私に受け止めきれるかな”

When I asked one of my friends she told me it’s along the lines “I’ve realized the sweet side Miura-kun has towards me and I wonder how I can get completely used to it.”

I understand that きれる can also mean completely so, is that correct?
Thank you very much.

Yes this 切れる is the potential form of 切る, which means "completely". See also: Is there any difference in meaning between 「抜け切る」 and 「抜ける」?

甘さを自覚した三浦くん means "Miura-kun who is aware of his (own) naiveness/optimism". This 甘さ is 考えの甘さ. The sentence is not saying "sweet side of Miura-kun" (三浦くんの甘さ). If you are still not good at relative clauses, please review its grammar now. 三浦くん is the subject of 自覚する.

I don't know how to translate 受け止める since there is not enough context. If this is a battle manga, this 受け止める should mean "to stop/endure his blow/attack". If this is a love story, it should mean something like "to accept him".

甘さを自覚した三浦くんは私に受け止めきれるかな。

I wonder if I can fully accept/stop Miura-kun, who is now aware of his (own) naiveness...

This is intended as a supplement to naruto's answer.

It might be easier to understand きれる via an easier example first. Take the verb 食べる (to eat). First attach きる to the base (continuative form /連用形) of the verb to make the verb:

食べきる to eat it all, eat everything until it is finished (definition)

Now you can make the potential form of 食べきる (godan) by changing it to:

食べきれる to able to eat it all, be able to eat everything until it is finished　　

As for 受け止めきれる, please refer to naruto's answer.