Python 2, 60 bytes

c=lambda f,n,l=[]:lambda a:n-1and c(f,n-1,l+[a])or f(*l+[a]) 

Try it online!

The footer is a tester which uses STDIN in the following way per line:

1. The function itself
2. The number of the function's arguments, ≥2
3. A list of the arguments ([a,b,...])

Note that, while a list of the arguments is given as input in the tester, in reality, the curried equivalent gets prepended to the list and the list is reduced by function call.

Python 2, 60 bytes

c=lambda f,n,l=[]:lambda a:n-1and c(f,n-1,l+[a])or f(*l+[a]) 

Try it online!

The footer is a tester which uses STDIN in the following way per line:

1. The function itself
2. The number of the function's arguments, ≥2
3. A list of the arguments ([a,b,...])

Note that, while a list of the arguments is given as input in the tester, in reality, the curried equivalent gets prepended to the list and the list is reduced by function call.

A similar 55-byte version has been kindly provided by ovs:

c=lambda f,n,*l:lambda a:n-1and c(f,n-1,*l,a)or f(*l,a) 

Try it online!