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ABSTRACT Feedback from active galactic nuclei and stellar processes changes the matter distribution on small scales, leading to significant systematic uncertainty in weak lensing constraints on cosmology. We investigate how the observable properties of group-scale haloes can constrain feedback’s impact on the matter distribution using Cosmology and Astrophysics with MachinE Learning Simulations (CAMELS). Extending the results of previous work to smaller halo masses and higher wavenumber, k, we find that the baryon fraction in haloes contains significant information about the impact of feedback on the matter power spectrum. We explore how the thermal Sunyaev Zel’dovich (tSZ) signal from group-scale haloes contains similar information. Using recent Dark Energy Survey weak lensing and Atacama Cosmology Telescope tSZ cross-correlation measurements and models trained on CAMELS, we obtain 10 per cent constraints on feedback effects on the power spectrum at $$k \sim 5\, h\, {\rm Mpc}^{-1}$$. We show that with future surveys, it will be possible to constrain baryonic effects on the power spectrum to $$\mathcal {O}(\lt 1~{{\ \rm per\ cent}})$$ at $$k = 1\, h\, {\rm Mpc}^{-1}$$ and $$\mathcal {O}(3~{{\ \rm per\ cent}})$$ at $$k = 5\, h\, {\rm Mpc}^{-1}$$ using the methods that we introduce here. Finally, we investigate the impact of feedback on the matter bispectrum, finding that tSZ observables are highly informative in this case.