To my questionable understanding;
A field is mediated by its particle representation, therefore, the Higgs field is mediated through an exchange with the Higgs boson.
I think technically speaking, the individual properties do not define the elementarity of the elementary particles, and therefore it is not required to have a field for every property, however, if a function of space and time has properties of energy, momentum, and force associated with it the definition for a field is fulfilled.