The short cycling is certainly a classic symptom of lack of air in the Hydrophore/ Pneupress tank.
Reading this I would like to point out the reason you want to add air to the bladder is to inflate it to its full size, so you may have to drain some of the water out before putting air in. So long as the bladder is expanding or contracting freely the air pressure inside the bladder and the water pressure outside will be the same and you have no way of telling what the volume of the bladder is. If the air pressure is slightly higher than the water pressure you know that the bladder must be full, and the air pressure is elastically stretching the bladder. Conversely, if the air pressure in the bladder is LESS than the water pressure you know that the bladder is squashed flatter than a pancake, either way you know it is not leaking.
Unfortunately, if you drained all the water out of the Hydrophore and pumped up the bladder to 18 psi you would probably burst the bladder!!
So stop the pump and slowly let the water go up to the accommodation, or run a tap somewhere, and keep adding air to maintain the bladder air pressure at 18 psi, when the water pressure falls below 18 psi you have the right volume of air, at the right pressure, inside the bladder and you can make fast the air filling line and switch the pump back to auto, oh and better shut that tap you were running. Everything is then set up correctly.
If you do not have a compressed air line with a pressure gauge for charging the bladder, the easy way is to drain the water out and inflate the bladder to its full size. Remember that if the bladder is full of air at Atmospheric Pressure then at 10 bar it will be compressed to 1/10 of its original volume! However do not put 10 bar air pressure on the bladder without a balancing water pressure on it!! Ideally you want the bladder fully expanded at or just below the pressure at which the pump cuts in. This will give the maximum quantity of water that can be delivered before the pump cuts in again. Reducing wear on the pump, motor and starter.
That tallies with your 18 psi air pressure in the bladder, 20 psi cut in, 40 psi cut out, so long as your gauges are accurate and the gauge lines are not blocked that should be fine.
As previously mentioned, an efficient non return or clack valve should be fitted in the pump discharge line.
With the reciprocating pump an accumulator vessel would have been fitted either as part of the pump or immediately after it to reduce pressure pulsations in the line. Usually the reciprocating pumps had a little "snifter cock" on the suction side that drew in air to keep this accumulator charged with air. (any surplus air was discharged with the water flow. This prevented pressure pulsations which would damage the pump and pipeline.
Another possible cause of short cycling could be pressure pulses in the pipe line, the pump starting could cause a pressure pulse to be transmitted down the pipe line and the pulse could be large enough to operate the pressure switch, these pressure pulses can also be reflected from tight bends etc so that there can be a whole series of them as a result of starting or stopping a pump. I notice several 90 degree bends after the pump. The simplest cure for this is to increase the length and reduce the diameter of the pipe between the pressure switch and the pump discharge pipe. Alternatively, connect the pressure switch to the Hydrophore tank itself, or to the Hydrophore OUTLET pipe. Or fit a pulsation damping valve similar to the ones used on Pressure gauges which are initially opened fully, resulting in big oscillations of the pressure reading and then slowly throttled in until the needle just trembles slightly, but that is difficult to do unless you have a pressure gauge next to the switch, and if the throttling valve became choked then the pressure switch would not work. Alternatively, rather than just damping the pressure pulses at the switch you could fit an accumulator between the pump and the switch, as in the old reciprocating pumps, that would stop any pressure pulses damaging the pump and pipes as well.
Another possible cause could be if the new pump had far too great a capacity causing a big pressure drop between the pump and the hydrophore, so the pressure at the pump would be almost immediately high enough to stop the pump, before the pressure in the Hydrophore had risen.
One ship I sailed on, the Kommandor Subsea, built by North Sea Shipyards in Denmark, had a very clever system, The tank did not have a bladder but a direct air/water interface. A level switch was fitted 1/3 below the top of the tank and a pressure switch mounted on the top. When the pressure fell to the cut in point the pressure switch sent power to the level switch, if the level switch was "UP" that in turn energised a solenoid valve that admitted air to the tank. If the level switch was "Down" it energised the starting coil for the water pump. Never seen that system before or since, but very clever, and very simple.
Hope some of this helps someone.
It is always better to ask a stupid question than to do a stupid thing.