AutoGyro Cavalon pilot reported a composite propeller blade failed and separated in flight and subsequently the engine died. The reporter landed safely at the departure airport.

Narrative

I'm a pilot of an AutoGyro Cavalon gyroplane; which is certified under the primary type category. On this flight we took off with no doors; and retained the hardware to mount the doors in the hangar. On the day of the incident; ATIS was reporting clear VMC conditions; with approximately 11-kt wind down the runway. While we were getting fuel I observed Aircraft Y rocking its wings on takeoff at about 200 feet; which suggested that there may be more turbulence farther above the runway. Before the flight; we picked up 12 gallons of fuel; did a final safety walk around the aircraft; taxied to the run-up; did our run-up checks - everything was normal - and then we were cleared to the runway for takeoff and a right downwind departure.Once I was aligned with the runway; I set the rotor brake selector switch to flight; brought the engine up to approximately 2000 RPM; and started to pre-rotate the main rotor; slowly advancing the engine power to keep the engine above 2000 RPM. When the rotor reached 200 RPM; I let go of the pre-rotator button; slowly pulled the cycling full aft and slowly advanced the throttle to approximately 4700 RPM. Once the nose started to lift I released pressure on the cyclic; brought the throttle to max; and let the aircraft keep accelerating till it started to fly. I then kept the aircraft close to the runway until it reached approximately 70 kt. and started to climb away from the runway. On the climb-out I remember significant side gusts entering the cabin but I kept the nose aligned with the yaw string on the windshield.Shortly before turning upwind to crosswind we heard a pop sound but all aircraft indications still looked good. We started to turn crosswind and elected to continue flying the pattern and monitor the engine. At the time I thought the pop was associated with a gust. Once we reached down wind at approximately 1000 feet we heard a second louder pop sound and then the whole aircraft began to shake violently. I immediately turned to the runway and [requested priority handling]. My first concern was that there was a failure in the rotor head but after I realized that the aircraft still responded to my cyclic inputs I felt relieved. Once I realized we were going to make the runway; I glanced at engine indications and saw that my engine RPM dropped to 0 and the vibrations damped down a bit. At 600 feet; I ran a mental trade between landing in the grass (no need for steep turns at low altitude; no possibility of colliding with an aircraft on takeoff) and runway (smoother surface in case of a bounced landing / less possibility of rollover and rotor impact; especially since I'm landing with an 11-kt tailwind) but once I heard the Tower clear the runway for me I picked the runway. At 200 feet I was confident I was going to make the runway so I started my turn to line up with it - I did have to slip a bit to correct for the crosswind from the north. We did end up touching down smoothly - ironically probably my smoothest gyrocopter landing to date.Once we stopped I tried to put on the rotor brake but realized that my main rotor was still spinning. Furthermore; I didn't know what damage the rotor may have sustained in flight and if its critical shaft modes have shifted due to the incident - causing us to vibrate and pitch over as it was spinning down. I made sure that I and my passenger stayed buckled in until that rotor came to a complete stop. At this point my passenger also reported smelling smoke so I had to stay alert for a possible fire and come up with a plan how to escape under the spinning rotors if the fire was detected. Throughout this period Tower was trying to communicate with me and directed ground crew to assist but because I was worried it wasn't over yet; I couldn't pay attention to all of the questions. Thankfully there was no fire; the rotors took about 5 minutes to stop and we were able to exit the aircraft.Once on the ground we saw that we were missing one of the composite blades on our propeller. It appeared to delaminate at the flange interface - the metallic flange is still there but all the fibers ripped off. The lost blade was installed just a few months prior because the original blade was discovered with a nick on the leading edge. The engine cowling and the tail was also covered in orange coolant from our aircraft. The ground crew also reported that a gray sweater fell out of the aircraft and landed on the runway during the initial climb-out. The sweater was initially lodged behind the seat and likely blew out during one of the side gusts during the climb-out. The ground crew at ZZZ got approval to relocate the gyrocopter off the runway. When we got back to the hangar and opened the engine cowling; we also saw that the rubber hoses connecting the carbonator and turbo to the engine got separated during the engine vibration. Thus; the vibration likely undid a bunch of hoses in the engine bay; killing the engine; and reduced the vibration.