Autogyro pilot reported a near miss with an aircraft in the traffic pattern after tower directed the autogyro to turn which put them in conflict with the aircraft.

Narrative

I was cleared for takeoff on runway XX takeoff with a northeast left downwind departure and advise 'on the go'. A 172 took off in front of me. On climb out on departure leg I confirmed 'Left Downwind Departure' with the tower then I turned left crosswind shortly after the end of the runway.I did realize that the 172 was staying in the pattern as it seem to be far away; then I turned on a close downwind.On my downwind leg the tower pointed out the 172 which was on a parallel track to me in a wide downwind leg. I acknowledged the 172 as 'insight' to the tower and the 172 acknowledged me as 'insight' to the tower.Then as I was proceeding northeast bound I was advised of landing traffic at my 11 oclock; then advised 'hard right; do it now' to avoid landing traffic which I flew over with possibly 200 to 300 feet separation. The towers radar was offline. The command to turn hard right actually put me over of the path of landing traffic; the separation would have been better without the hard right command.Being a rotorcraft; I generally fly a tighter pattern if no others are in the pattern. My error was not realizing the 172 was staying in the pattern. If I had followed the 172; the timing would have been perfect; and this would have been a non-issue. With this incident it has become clear to me that my situational awareness of traffic in the pattern needs improvement.