We present a framework utilizing higher-order cross-correlations to study neutral hydrogen ( \(\textrm{HI}\) ) clustering around galaxies, using data from the IllustrisTNG300 simulation. This method computes the joint distributions of k-nearest neighbor ( \(k\mathrm{{NN}}\) ) galaxies and the \(\textrm{HI}\) brightness temperature field smoothed at relevant scales, providing sensitivity to all higher-order correlations. By adding thermal noise and applying foreground cleaning, we detect \(\textrm{HI}\) -galaxy cross-correlations at \({>}30\sigma \) across \(r = [3, 12] \, h^{-1} \mathrm{{Mpc}}\) using the \(k\mathrm{{NN}}\) -field framework. In contrast, the two-point correlation function ( \(2\mathrm{{PCF}}\) ) detection depends on the foreground filter, showing a spurious \(8\sigma \) detection with a sharp filter. These results demonstrate the enhanced constraining power of the \(k\mathrm{{NN}}\) -field framework.