Weak gravitational lensing is one of the most powerful tools for cosmology, while subject to challenges in quantifying subtle systematic biases. The Point Spread Function (PSF) can cause biases in weak lensing shear inference when the PSF model does not match the true PSF that is convolved with the galaxy light profile. Although the effect of PSF size and shape errors - i.e., errors in second moments - is well studied, weak lensing systematics associated with errors in higher moments of the PSF model require further investigation. The goal of our study is to estimate their potential impact for LSST weak lensing analysis. We go beyond second moments of the PSF by using image simulations to relate multiplicative bias in shear to errors in the higher moments of the PSF model. We find that the current level of errors in higher moments of the PSF model in data from the Hyper Suprime-Cam (HSC) survey can induce a $\sim 0.05 $ per cent shear bias, making this effect unimportant for ongoing surveys but relevant at the precision of upcoming surveys such as LSST.