'With that that thing that beats beneath my skin'
^For starters, you only need one that in the beginning of the line, not two. Secondly, thing is too vague, I undersatnd youre probably talking about your heart, but thing is not in any way poetic.

'It's been tattered torn and smothered'
^Put a comma after tattered, it will flow better with that pause in there

'Taking me many a month'
^I didnt understand this line? Did you mean to say 'Taking me many months'?

'And the pain burned quite sweetly'
^I dont think and is necessary here.. it flows better without. Loved pain burned quite sweetly, never thought of something quite like that before or seen something like that.

A good poem, some things need some altering bother than that a nice poem.