AN LJ35 CREW; TAXIING FOR TKOF AT HPN; CROSSED THEIR DEP RWY WITHOUT A CLRNC DURING A NIGHT OP.

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AT HPN TAXIING AN LR35 TO RWY 34. CAPT ASKED CLRNC TO TAXI RWY 34 OUT OF LOCATION X AND GND CTLR REPLIED; 'CLRED TO CROSS RWY 29; TAXI RWY 34.' CLRLY; THE CTLR THOUGHT WE WERE ON N END OF THE FIELD AT TXWY A. PROB IS THE CAPT REPLIED 'CLRED CROSS RWY 29; TAXI RWY 34.' I WAS CONTINUING DOWN CHKLIST WITH CAPT RESPONDING TO CHALLENGE. I THEN NOTICED WE WERE ABOUT TO CROSS RWY 34 AT TXWY K. I SAID 'STOP; WE ARE NOT CLRED TO CROSS RWY 34; ONLY RWY 29.' HE THOUGHT WE WERE XING RWY 29. WE HAD ALREADY CROSSED HOLD SHORT LINE; BUT NOT ON ACTUAL ACTIVE RWY. HE CALLED GND AND ASKED FOR CLRNC AGAIN AND IF WE WERE CLRED TO CROSS RWY 34. CTLR REPLIED WITH ORIGINAL CLRNC -- CAPT READING BACK CLRED TO CROSS RWY 34. HE THEN CROSSED RWY 34; EVEN THOUGH I TOLD HIM WE WERE STILL NOT CLRED TO CROSS. HE WAS CONVINCED IT WAS STILL RWY 29; UNTIL WE WERE ON AND XING RWY 34. GND CTLR WAS NOT AWARE WHERE WE WERE AT; THUS GIVING US WRONG TAXI INSTRUCTIONS. THAT; AND CAPT'S LACK OF AWARENESS CAUSED US TO CROSS RWY 34 AT TXWY K WITHOUT CLRNC. I INFORMED MY CHIEF PLT; AND HE IS TAKING CORRECTIVE ACTION TO INFORM ALL PLTS AS TO THE IMPORTANT VIGILANCE AND COM NEEDED FOR SAFE OPS.