Cycling just means that you're using less water than the pump can supply. It could be that it's higher flow than the previous pump, or because the flow is limited by the faucet valve not being open all the way, the aerator being partly plugged, or what have you.
An accumulator tank will not prevent cycling, but will make the period longer and thus less annoying (i.e. several seconds on, several off, rather than once per second). It will also smooth out the pulsations in the water flow some, or at least make them less obviously noticeable.