A Center Controller reported they had difficulty communicating with an aircraft requesting special handling due to fuel leak. The reporter stated their radio coverage is not sufficient for the size of the sector.

Narrative

Aircraft X [requested priority handling] due to emergency fuel levels. Later on they would say they were checking for a possible fuel leak. I immediately cleared them directly to their original destination per their request. Later on when I was trying to gather the necessary information I needed to help the [priority handling] aircraft; I kept getting blocked on frequency from pilots who kept trying to check in on frequency. The problem is that they were out of range of the frequency coverage; so I could not hear them calling. After that Aircraft X reported that they were also getting blocked from unknown aircraft. I then had another pilot relay on frequency telling everyone to standby because I had a [priority handling] aircraft. When I had control of the frequency I then was requesting information from Aircraft X and they were coming in extremely broken; even though they were no more than 90 miles from the frequency site. Eventually they diverted to ZZZ1 and landed safely. Later on; a pilot that was also in my frequency coverage stated that my transmissions were coming in very broken. The airspace is not designed to have a frequency of this range; or of this quality. It is too big. Aircraft should be reporting further south when they are within range of the frequency. This frequency also needs to be fixed because it is having a lot of issues within the 'known' frequency coverage. The current state of the sector is unsafe due to us not being able to service emergency traffic quickly and efficiently to make sure they get the assistance needed in a stressful time. They do not have time to keep repeating themselves over and over again when they should be focusing on flying the airplane. It also adds a lot of extra workload for the controllers; if we have to keep repeating ourselves it gives us less time to come up with a safe plan for the aircraft in need.