A bladder tank fault won't keep the pump on. You need to fix this....your electric bill will be a bit higher that expected with the pump running all of the time.
Sounds like there is a leak (you didn't mention running out of water, only that the pressure was low), since it can't build up proper pressure. A pressure tank only holds pressure between pump cycles, it doesn't make pressure, it stores energy (pressure) from the pump when it is running.
One of the guys that deal with pumps will have some idea how to diagnose this. In the meantime, you probably should shut the pump power off until needed.