Gramatically speaking both are correct. 

Naturally, I may say 'Hablarme' instead of 'hablarme a mi'. 

 'Hablarme a mi' is redundant. You don't really need to add 'a mi' because with the ending 'me' you already say who is receiving the action. Same is applied with 'hablarte', 'hablarle', 'hablarles', and so on. 

Here are some examples:

-Ella no va a hablarme a mi. (She is not talking to me).

-Deberías dejar de hablarle a ella. (You should stop talking to her).

(I am native spanish speaker willing to practice my english. If you want, message me and I can help you practice spanish.)

I think that the correct way to write it will be: 

¿Puedes hablarme de eso? (without "a mí") 

When you say "hablarme" you don't need to write "a mí" because the conjugation form already says who's the subject who receives the action verb "hablar". 

The same in the other hand... when you say:

 "No va a hablarme otra vez"

It's obvious that the one that it's receiving the action is "you".

Summarizing: when you add the ending "-me"  to the verb "hablar" [hablarme] you don't need to say "a mí" because it is repetitive. 

I hope this can be clear for you. Greattings from Mexico. 

Hablame, is when you are actually talking to that person, Hablarme is just depends on the context and most like is a question

like some common sentences in English

Talk to me about it..  (hablame de eso) or talk to me (hablame)

or can you talk to me? (puedes hablarme?)