Dialectic differences.

  Yorkshire dialect word     : barn
  Generally accepted meaning : child
                              (especially a young child, infant) 	
  Old Norse source word      : barn
                               Same as bairn, which comes from
                               the Old English bearn.  bairn is
                               used as an alternative in some
                               parts of Yorkshire, the other
                               northern counties and Scotland.

From http://www.viking.no/e/england/yorkshire_norse.htm

Pretty sure 'barn' in this sense means 'bound', as in 'heading' or 'going'. "Hey up, Young Pokey. Is tha barn darn t' pit?" reads to me as "Hello, son of Poke. Are you heading down the pit?"

Given that this conversation appears to be directed at the child in question, and the child responds by saying "Ah'm barn darn t' thutty-niners", seems unlikely he'd refer to himself as a 'barn'. Makes more sense to me that he's agreeing - he's bound down the thirty-niners.