We present various recent updates related to the MSHT20 PDF sets, and particularly to the MSHT20 approximate N$^3$LO (aN$^3$LO) PDFs. These include an investigation of including either inclusive jet data or dijet data, the detailed treatment of $Z\,p_T$ data, and the impact on the gluon PDF in particular. We also very briefly summarize the determination of the strong coupling via an aN$^3$LO global PDF fit, finding $\alpha_S(M_Z^2)=0.1170\pm 0.016$, and of including QED corrections in the aN$^3$LO analysis. Finally we consider the preliminary finding associated with the effect of improved knowledge of the N$^3$LO splitting functions on the aN$^3$LO PDFs. In particular we note a moderate impact on the gluon distribution, i.e. within the uncertainty, and a more limited impact on the gluon-gluon luminosity unless strong rapidity cuts are imposed. Hence, we conclude that while details of our aN$^3$LO PDFs will change with more precise splitting functions, only relatively minor effects on predictions are likely to result.