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We study the first-order relativistic correction to the associated productionof $J/\psi$ with light hadrons at $B$ factory experiments at $\sqrt{s}=10.58$GeV, in the context of NRQCD factorization. We employ a strategy for NRQCDexpansion that slightly deviates from the orthodox doctrine, in that thematching coefficients are not truly of ``short-distance" nature, but explicitlydepend upon physical kinematic variables rather than partonic ones. Ourmatching method, with validity guaranteed by the Gremm-Kapustin relation, isparticularly suited for the inclusive quarkonium production and decay processeswith involved kinematics, exemplified by the process $e^+e^-\to J/\psi+gg$considered in this work. Despite some intrinsic ambiguity affiliated with theorder-$v^2$ NRQCD matrix element, if we choose its value as what has beenextracted from a recent Cornell-potential-model-based analysis, including therelative order-$v^2$ effect is found to increase the lowest-order predictionfor the integrated $J/\psi$ cross section by about 30\%, and exert a modestimpact on $J/\psi$ energy, angular and polarization distributions except nearthe very upper end of the $J/\psi$ energy. The order-$v^2$ contribution to theenergy spectrum becomes logarithmically divergent at the maximum of $J/\psi$energy. A consistent analysis may require that these large end-point logarithmsbe resummed to all orders in $\alpha_s$.

Color-singlet relativistic correction to inclusive $J/\psi$ production associated with light hadrons at $B$ factories