The superimposed pilot approach is widely used in OFDM because of its higher spectral efficiency. However, transmission of pilots and data in the same resource elements causes pilot-data contamination, degrading estimation and detection performance. To address these issues, we propose a non-iterative channel-refinement receiver design using a convolutional neural network (CNN) for single-input, single-output (SISO) OFDM. The proposed work uses a sparse diamond-shaped pilot pattern. The pilot pattern is superimposed with data symbols only at specific resource elements, restricting pilot-data interference to a limited number of resource elements. The proposed channel refinement approach comprises two steps: initial channel estimation using a least-squares (LS) estimator in the presence of pilot-data interference, followed by refinement of estimated channel coefficients using a lightweight CNN, trained under various time-varying channel conditions.  By learning the mapping from coarse channel estimates obtained in the presence of pilot-data interference to the corresponding true channel responses, the trained CNN refines the initial coarse channel estimation. The simulation results verify significantly improved estimation accuracy and corresponding data detection performance for the proposed CNN-based channel refinement approach, which outperforms conventional interference refinement approaches.