What I meant was that it would be unusual for two separate tubes to get blocked at the same time, wouldn't it?
Maybe, but if the 'gunge' was some sort of growth, or other build up, it is possible it has affected everything equally. Seen it happen with 'splitter' laundry taps at my parents place - caused flow problems to tub and washing machine.
What you describe, about it taking a long squirt or several squirts sometimes, then getting a powerful squirt... it could be blockage in the pipe (as suggested), could also be air getting into the system (split/perished fluid pipes - however than would mean both front and rear pipes at the same time...), might be an airlock... but I would expect it either to cause no flow or be cleared by the water being pumped though (being an open to air ended system where, at least the front, nozzles are probably the high-point of the pipe and not a large volume of water being stored in the pipes), as you say it could be the pump.
What
eye30 advises, about blowing air through both sections of pipe for the front nozzles (once disconnecting the join) and the water in the fluid tank bubbling - might have worked if there was stuff in pipes and/or pump (but I would assume it would have got trapped by the filter on the pump). Maybe by doing so he cleared the pipe of all water, and so any airlocks (if present). Suppose it is worth a shot.
Though I would be more tempted to disconnect at a join (as indicated by
eye30) then run the pump and see what happens... strong flow always? same symptoms? If same disconnect the pipes at the pump and try (careful not to get stuff wet) - if still the same could be pump needs replacing (or maybe clearing by air - I would be tempted to do so without the filter in place, or clear it before running pump again). Otherwise it indicates the pipes... which, per
eye30 advice might be cleared by blowing air through. At the very least you will probably be able to ID which section the problem lies in.