Cavity walls are just that, there will be air ingress into the cavity if it isn't filled with foam. If there's holes in the inner wall air will get into the house.
Often with fitted kitchen units, cold air will get in behind them from below the floor under the skirting board.
If there's no heat getting into the cupboard, it will always be cold. if as you say, you have an internal door to the space, the easiest solution would be to make sure that has draught excluder and forget how cold it is in there.
We have a large and small cupboard under the stairs. The large one contains my wife's "fleet" of vacuum cleaners plus the tumble dryer where the exhaust air exits through a vent I made in the air brick in there. As the dryer is "on all day" it never feels cold in there when I open the door.
The small one near the foot of the stairs has the gas meter and it's always cold in there, as there's just a scrap of lino on top of bare floorboards.