Ground-penetrating radar (GPR) is an effective geophysical-electromagnetic method for detecting subsurface targets. However, target echoes are often flooded by strong clutter, which affects the detection performance. Clutter suppression is an active area in GPR research, and the current trend is to use methods based on low-rank sparse decomposition (LRSD), which has significant advantages over traditional low-rank methods. In recent years, the tensor robust principal component analysis (TRPCA) method has been proposed for use in GPR clutter suppression. Although good results have been achieved, TRPCA does not retain the information of the low-rank part well by shrinking all the singular values equally. There is a significant gap between all singular values in B-scan data in practical applications; therefore, different singular values should be treated differently. Therefore, this study proposes a TRPCA method (Schatten p-TRPCA, p-TRPCA) based on weighted tensor Schatten p-paradigm minimisation, which assigns different weights to different singular values and decomposes the B-scan image into a low-rank clutter matrix and a sparse target matrix. The method was evaluated using simulation experiments and real test data, and its performance was found to be better than those of TRPCA and traditional methods.