We present a search for a new heavy particle X produced in association with a top quark, p (p) over bar -> t(X -> (t) over barq) or p (p) over bar -> (t) over bar((X) over bar -> t (q) over bar), where q stands for up quarks and down quarks. Such a particle may explain the recent anomalous measurements of top-quark forward-backward asymmetry. If the light-flavor quark (q) is reconstructed as a jet (j), this gives a (t) over bar + j or t + j resonance in t (t) over bar + jet events, a previously unexplored experimental signature. In a sample of events with exactly one lepton, missing transverse momentum and at least five jets, corresponding to an integrated luminosity of 8.7 fb(-1) collected by the CDF II detector, we find the data to be consistent with the standard model. We set cross-section upper limits on the production (p (p) over bar -> Xt or (X) over bar(t) over bar) at 95% confidence level from 0.61 pb to 0.02 pb for X masses ranging from 200 GeV/c(2) to 800 GeV/c(2), respectively. RI song, hao/I-2782-2012