I'm guessing that your question is this: What is the difference between:
It's late. I've got to go home.
It's late. I got to go home.
The answer is simple - the first is correct, informal spoken English. The second is lazy, incorrect English.
In fact, both are versions of the construction:
It's late. I have to go home. (Tenho que ir para casa - just like Portuguese)
The verb 'have' is quite a weak sound. As you know, it tends to contract to 've. 'I've to go' doesn't sound good, so we added the word 'got' to make it sound more complete - 'I've got to go'. Then the "ve" disappeared completely, and we are left with 'I got to go' (sometimes - even more incorrectly - written as '"I gotta go"). So in fact we have four versions of the same expression:
1) I have to go.
2) I've got to go.
3) I got to go
4) I gotta go.
(1) is the most correct/formal.
(2) is fine for speaking and informal writing
(3) and (4) are ok to say, but don't write them.
I hope that answers your question.