A MOONEY M20J PLT CROSSES THE FINAL APCH COURSE FOR THE RWY THAT HE HAD JUST DEPARTED FROM; COMING WITHIN APPROX 500 FT OF TWO APCHING ACFT. THE MOONEY PLT KNEW ABOUT THE INBOUND ACFT BEFOREHAND.

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I DEPARTED RWY 16 FROM HPN AND WAS GIVEN A R TURN. I HAD TOLD THE TWR THAT I WAS GOING TO POU WHICH USUALLY WOULD BE A L TURN OUT TO AVOID RWY 16 APCH. I WAS TOLD TO PROCEED 'ON COURSE' WHICH WAS A 360 DEG HDG AND WOULD TAKE ME ACROSS THE APCH COURSE OF RWY 16. I WAS TOLD BY TWR 'FREQ CHANGE APPROVED' I VOLUNTARILY SWITCHED TO 126.40 (NY APCH) FOR TA'S. THE CTLR WAS QUITE BUSY AND I HEARD HIM CLR 2 ACFT (1 JET AND 1 BEECH 1900) TO INTERCEPT THE LOC FOR RWY 16 AT HPN. AFTER SEVERAL TRIES I REACHED HIM FOR ADVISORIES. I SAW AND AVOIDED BOTH ACFT ON THEIR APCHS. HE TOLD ME IT WAS NOT A GOOD IDEA TO CROSS THE APCH END OF A RWY (WHICH I WAS DOING); AND TOLD ME HE ALMOST HAD TO CANCEL LNDG BECAUSE OF MY PRESENCE. WHEN TWR ADVISED ME TO 'ON COURSE' I ASKED IF I WAS CLR OF THE APCH TO RWY 16 AND THEY SAID 'FREQ CHANGE BECAUSE OF THE NUMBER OF LNDG AND TKOF AT THE TIME;' IT APPEARED THAT TWR WAS BUSY AND WASN'T TRACKING ALL DEPARTING ACFT; 2 RWYS WERE BEING USED. I WOULD THINK THAT THEY SHOULD HAVE INSTRUCTED ME TO 'OVERHEAD THE ARPT AT 2500 FT' AND THEN HAVE ME GO 'ON COURSE.' I WAS VERY AWARE OF THE OTHER ACFT; BUT UNTIL I WAS IN NY APCH (126.4) RADAR AND IDENTED I CAN SEE THAT IT WAS A PROB. NY WAS VERY NICE AND WE DISCUSSED THE SIT AND HE SAID 'SOMETIMES HPN'S IDEAS OF WHERE TO PUT PEOPLE ARE DIFFERENT THAN NY'S;' OR WORD TO THAT MATTER. HAD I NOT BEEN ALERT AND TALKING TO NY AS SOON AS I DID; THERE COULD HAVE BEEN A COUPLE OF 'BIG' INCIDENTS. NY APCH SHOULD INFORM HPN OF DESIRED DEP RTES DURING OPS OF MARGINAL VFR AND VFR; TO AVOID THESE SITS.