  Starting from a complete set of relativistic nucleon-nucleon contact
operators up to order $O(p^4)$ of the expansion in the soft (relative or
nucleon) momentum $p$, we show that non-relativistic expansions of relativistic
operators involve twenty-six independent combinations, two starting at
$O(p^0)$, seven at order $O(p^2)$ and seventeen at order $O(p^4)$. This
demonstrates the existence of two low-energy free constants that parameterize
interactions dependent on the total momentum of the pair of nucleons $P$. The
latter, through the use of a unitary transformation, can be removed in the
two-nucleon fourth-order contact interaction of the Chiral Effective Field
Theory, generating a three-nucleon interaction at the same order. Within a
hybrid approach in which this interaction is considered together with the
phenomenological potential AV18, we show that the LECs involved can be used to
fit very accurate data on the polarization observables of the low-energy $p-d$
scattering, in particular the $A_y$ asymmetry.
