Neuromorphic computing, utilizing memristor crossbar arrays offers a promising solution for edge device by enabling cognitive tasks such as pattern recognition, thanks to its small size, low power consumption and in-memory processing capability. Among various memristor architectures for neuromorphic computing, the single crossbar stands out due to its efficiency in terms of power and area saving when performing the pattern recognition task based on the Exclusive-NOR operation. However, this architecture exhibits performance instability when handling sparse data. To overcome this limitation, this work proposes an enhanced single crossbar architecture that integrates a column-wise constant current generating circuit to enable the full-XNOR functionality. This design preserves the advantages of the baseline single crossbar design while substantially improving robustness to sparse data. Simulation results, evaluated with images of varying data densities (0.25, 0.5, and 0.75), demonstrate the addition of the constant term elevates the column currents uniformly, allowing the Winner-take-all circuit to function efficiently and produce correct outputs. When the data density drops below 0.3, the proposed single memristor crossbar with a column-wise constant term maintains a 100% recognition rate, whereas the architecture without the constant term experiences significant degradation, with recognition rate dropping to as low as 0%. The proposed single memristor crossbar architecture with constant column-wise term overcomes the limitation of the baseline single crossbar design introduced in previous study, while retaining its key advantages. Specifically, it reduces the number of memristors by 50%, and achieves recognition accuracy improvements of 7% and 4% under a 10% device defects rate, and 11.4% and 3.3% under the 40% memristance variation, compared to the complementary and the twin crossbar architectures, respectively.