Neutral hydrogen (\(\rm{HI}\)) \(21\)-cm intensity mapping (IM) offers an efficient technique for mapping the large-scale structures in the universe. We introduce the 'Cross' Tapered Gridded Estimator (Cross TGE), which cross-correlates two cross-polarizations (RR and LL) to estimate the multi-frequency angular power spectrum (MAPS) \(C_{\ell}(\Delta\nu)\). We expect this to mitigate several effects like noise bias, calibration errors etc., which affect the 'Total' TGE which combines the two polarizations. Here we apply the Cross TGE on a \(24.4 \,\rm{MHz}\) bandwidth uGMRT Band \(3\) data centred at \(432.8 \,\rm{MHz}\) aiming \(\rm{HI}\) IM at \(z=2.28\). The measured \(C_{\ell}(\Delta\nu)\) is modelled to yield maximum likelihood estimates of the foregrounds and the spherical power spectrum \(P(k)\) in several \(k\) bins. Considering the mean squared brightness temperature fluctuations, we report a \(2\sigma\) upper limit \(\Delta_{UL}^{2}(k) \le (58.67)^{2} \, {\rm mK}^{2}\) at \(k=0.804 \, {\rm Mpc}^{-1}\) which is a factor of \(5.2\) improvement on our previous estimate based on the Total TGE. Assuming that the \(\rm{HI}\) traces the underlying matter distribution, we have modelled \(C_{\ell}(\Delta\nu)\) to simultaneously estimate the foregrounds and \([\Omega_{\rm{HI}} b_{\rm{HI}}] \) where \(\Omega_{\rm{HI}}\) and \(b_{\rm{HI}}\) are the \(\rm{HI}\) density and linear bias parameters respectively. We obtain a best fit value of \([\Omega_{\rm{HI}}b_{\rm{HI}}]^2 = 7.51\times 10^{-4} \pm 1.47\times 10^{-3}\) which is consistent with noise. Although the \(2\sigma\) upper limit \([\Omega_{\rm{HI}}b_{\rm{HI}}]_{UL} \leq 0.061\) is \(\sim 50\) times larger than the expected value, this is a considerable improvement over earlier works at this redshift.