There are two issues here: the informal dropping of the 'have' of 'have got' (non-standard and arguably incorrect) and the reduced relative clause (standard and perfectly correct).
The standard way to say this would be this: "How many more mines HAVE you GOT buried out there?".
This is a slightly more informal way of saying "How many more mines DO you HAVE buried out there?". Both are grammatically correct.
However, the word 'have' usually gets shortened to 've' in spoken English ( as in 'I've got...), and it is sometimes dropped entirely, which is what this speaker has done here.
Note that the dropping the 'have' or 'has' is not seen as grammatically acceptable: young children often do this, and in an adult it's seen as a sign of being uneducated.
It's common to come across the dropped 'have' in songs and movies, but you shouldn't normally write English this way. In written English, it's only really acceptable in quoted dialogue.
As for the second part of the sentence with the reduced relative clause, this is perfectly normal in both written and spoken English. When we have a participle ( in this case, the past participle 'buried'), there is no need to use the full relative clause. Yes, the extended form of the sentence is "How many more mines have you got (which are) buried out there?", but we wouldn't usually say this. The reduced form is more concise and far more natural.