If you run it say for 10 min will it clear up and run just water? I suspect that it cycles because of the air and the compression and expansion of air in the line. In theory if you have no water leaks then I would say the air is coming in from the suction side of your pump... and all the suggestions above point to this.
We have had this problem a few times. Typically it is the outside shower. As we spool up the hose and shove it into the small compartment we somehow, all so slightly turn the hot water valve, and it does not drip out of the showerhead, or see any water in the compartment. On the other hand the filter screen gave us issues midway through the first year we had it. The housing was full of spider web cracks. I am guessing it was not winterized correctly. During the last campout with the issue being the worst I wrapped the cup in electrical tape and it almost cured the issue. Replaced the clear cup and the issue went away. The only other time we have had this issue was when the HTT was new. The thumb screws that connect the pipes to the bathroom facet were a little loose. We never felt any water on the fittings, but I tightened them up a few times and the pump stopped cycling.
If the pump is cycling you have a pressure leak somewhere. Start with the easy items and work your way to the hard areas.