A better reading of "Hey up, Young Pokey. Is tha barn darn t' pit?" would be "Hey up, Young Pokey. Are you going down the pit?".

    "Is tha" - "Are you" (from "thou")
    "barn darn" - Barn here mean "bound" or "going"; "darn" is just "down"
    " 'pit" - The pit (roughly the mine)

Similarly, "Aye, sither. Ah'm barn darn t' thutty-niners wi' t' fa'ther" would be read as "Aye, you see. I'm going down the pit with my father".

Slight aside: Anyone else from Yorkshire use/hear "seefing" to mean "seeing if" (as in "I was seefing/seeing if tickets were available"). I hear it relatively often, but I don't recall ever seeing it written, even in dialect.