Second-order variables: elements of domain only without quantifiers?

[nomadreid]

On one side one can define second-order variables as ranging over all elements of P^k(M) for all natural numbers k (P=power set of M, M is the universe of the model, superscript being iteration). On the other side it is sometimes defined as ranging over all first-order relations and predicates. In this latter definition, does "predicates" include first-order sentences with quantifiers and variables, or only first-order sentences with only constant symbols?
Thanks.

 

[nomadreid]

er, sorry, I meant P(M^k), where the exponent refers to the Cartesian product