Recent detections of the cross-correlation of the thermal Sunyaev–Zel ’ dovich ( tSZ ) effect and weak gravitational lensing ( WL ) enable unique studies of cluster astrophysics and cosmology . In this work , we present constraints on the amplitude of the non-thermal pressure fraction in galaxy clusters , \alpha _ { 0 } , and the amplitude of the matter power spectrum , \sigma _ { 8 } , using measurements of the tSZ power spectrum from Planck , and the tSZ-WL cross-correlation from Planck and the Red Cluster Sequence Lensing Survey . We fit the data to a semi-analytic model with the covariance matrix using N -body simulations . We find that the tSZ power spectrum alone prefers \sigma _ { 8 } \sim 0.85 and a large fraction of non-thermal pressure ( \alpha _ { 0 } \sim 0.2 – 0.3 ) . The tSZ-WL cross-correlation on the other hand prefers a significantly lower \sigma _ { 8 } \sim 0.6 , and low \alpha _ { 0 } \sim 0.05 . We show that this tension can be mitigated by allowing for a steep slope in the stellar-mass-halo-mass relation , which would cause a reduction of the gas in low-mass halos . In such a model , the combined data prefer \sigma _ { 8 } \sim 0.7 and \alpha _ { 0 } \sim 0.2 , consistent with predictions from hydrodynamical simulations .