  We consider heavy meson-antimeson pairs and their coupling to quarkonium in
the context of nonrelativistic EFTs incorporating the adiabatic expansion. We
work out all the leading order couplings of quarkonium to heavy meson-antimeson
pairs and obtain their contributions to the masses and widths of quarkonia. We
match the new potentials terms to NRQCD. Using the available lattice data for
the coupled system of quarkonium and the lowest laying heavy meson-antimeson
pair, we extract the mixing potential and use it to compute numerically the
contributions of $D\bar{D}(B\bar{B})$ and $D_s\bar{D}_s(B_s\bar{B}_s)$ to the
masses and widths of the charmonium (bottomonium) states for $l=0,1,2$ and up
to $n=6$ covering the states in threshold region. When a quarkonium state and a
heavy meson-antimeson pair are separated by small energy gaps, their
interactions can be described by a threshold EFT with contact interactions. We
work out the matching between the two EFTs obtaining the couplings of the
threshold EFT in terms of the mixing potential and quarkonium wave functions.
