Özür dilerim ama mesajları gönderemem

I´m sorry but I can´t send messages

I think it´s gönderemiyorum because you are referring to now. It´s confusing to me because the tense you used is sometimes called simple present.

I think it is mesaj, not sure why, but plural is used less often in Turkish.

Either way, I hope someone with more Turkish knowledge can help.

Özür dilerim ama mesaj gönderemem.

I think that you dont want to send messages to him right? If so, my translation is correct.

How would the meaning change with this: Özür dilerim ama mesaj gönderemiyorum

And why is message singular, because it could be one?

If it was gönderemiyorum, it would be because of a problem.

Think "mesaj göndermek" as one verb.  When you say "mesaj gönderemiyorum", you say that you are not able to do this action. So it doesnt matter that you want to send one or more messages. Because it will not change the meaning.

But if the sentence was like that: I cant send the messages.

Than translation would be like this: Mesajları gönderemiyorum.

Because we have a definite object now.