Hi seba
Ex: There is no furniture in the house except
for a fridge in the kitchen.
Ex: There is no furniture in the house except
for fridge in the kitchen.

=>
fridge is a count noun: it requires a determiner.
'a' = non-defined
'the' = defined
Ex: There is no furniture in the house except
for the fridge in the kitchen <define>
that we bought 20 years ago.
Ex: There is no furniture in the house except
for a fridge in the kitchen, <define>
the one that we bought 20 years ago.