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We present a sensitive search with the Karl G. Jansky Very Large Array (VLA) for the radio counterpart of the gravitational wave candidate S191216ap, classified as a binary black hole merger, and suggested to be a possible multi-messenger event, based on the detection of a high energy neutrino and a TeV photon. We carried out a blind search at C band (4--8 GHz) over 0.3 deg$^2$ of the gamma-ray counterpart of S191216ap reported by the High-Altitude Water Cherenkov Observatory (HAWC). Our search, spanning three epochs over 130 days post-merger and having mean source-detection threshold of 75$μ$Jy/beam (4$σ$), yielded 5 variable sources associated with AGN activity and no definitive counterpart of S191216ap. We find $<$2\% ($3.0\pm1.3$\%) of the persistent radio sources at 6 GHz to be variable on a timescale of $<$1 week (week--months), consistent with previous radio variability studies. Our 4$σ$ radio luminosity upper limit of $\sim$1.2$\times{10}^{28}$ erg s$^{-1}$ Hz$^{-1}$ on the afterglow of S191216ap, within the HAWC error region, is 5--10 times deeper than previous BBH radio afterglow searches. Comparing this upper limit with theoretical expectations given by \citeauthor{Perna2019} for putative jets launched by BBH mergers, for on-axis jets having energy $\simeq10^{49}$ erg, we can rule out jet opening angles $\lesssim$20 degrees (assuming that the counterpart lies within the 1$σ$ HAWC region that we observed).