Is it possible that the clay that holds the crown in place, can seep through into the gum and cause a slight sensation to the right side of my face?

Clay ?: Really not sure if you mean cement or what sensation you are experiencing following your dental treatment. Best bet is to have your dentist take a look.

No: More likely that you have an infection, or cement was-out, under the crown. Please see your Dentist for treatment before this gets out of hand.

It is possible: tooth decay under the crown. If decay under a crown is left too long, the tooth can decay enough that the tooth may not be fixable. Call your dentist.

See DDS: Have your dentist clinically and radiographically evaluate the tooth for possible decay and need for removal of decay and crown replacement.