Cold water comes in via 'city water' connection OR from on-board pump discharge. Feeds ALL cold water lines to fixtures and to inlet of water heater.....which then supplies all hot water. The two systems spit at water heater.
Having good hot water pressure and low cold water pressure is very strange as the cold water pressure is what forces water thru water heater to get hot water.
Typically low flow is due to clogged aerators on faucets but that affects both hot & cold.
Low hot water pressure s typically bad check valve on water heater outlet.
Can't think of a reason for good hot and low cold water pressure :H
Basic water system.....

Problem (low cold water pressure) happen with either water source (city water...hose and/or on-board pump)????