Recent cosmological observations have achieved high-precision measurements of the Universe's expansion history, prompting the use of nonparametric methods such as Gaussian processes (GP) regression. We apply GP regression for reconstructing the Hubble parameter using CC data, with improved covariance modeling and latest study in CC data. In addition, we perform a joint analysis combining BAO, SN Ia, and a CMB prior, which constitutes the standard and most constraining framework in cosmology. By comparing reconstructions in redshift space z and transformed space log(z+1), we evaluate six kernel functions using nested sampling (NS) and approximate Bayesian computation rejection (ABC rejection) methods and analyze the construction of Hubble constant H0 in different models. Our analysis demonstrates that reconstructions in log(z+1) space remain physically reasonable, offering a viable alternative to conventional z space approaches, while the introduction of non-diagonal covariance matrices in CC data leads to degraded reconstruction quality, suggesting that simplified diagonal forms may be preferable for reconstruction. And we find robust evidence for an observable deviation from the ΛCDM model under the joint constraints of BAO and SNe Ia data with a CMB prior. These findings underscore the importance of task-specific kernel selection in GP-based cosmological inference. In particular, our findings suggest that careful preliminary screening of kernel functions, based on the physical quantities of interest, is essential for reliable inference in cosmological research using GP.