Using Subaru Hyper Suprime-Cam (HSC) year 1 data, we perform the first $k$-cut cosmic shear analysis constraining both $Lambda$CDM and $f(R)$ Hu-Sawicki modified gravity. To generate the $f(R)$ cosmic shear theory vector, we use the matter power spectrum emulator trained on COLA (COmoving Lagrangian Acceleration) simulations. The $k$-cut method is used to significantly down-weight sensitivity to small scale ($k > 1 h {rm Mpc }^{-1}$) modes in the matter power spectrum where the emulator is less accurate, while simultaneously ensuring our results are robust to baryonic feedback model uncertainty. We have also developed a test to ensure that the effects of poorly modeled small scales are nulled as intended. For $Lambda$CDM we find $S_8 = sigma_8 (Omega_m / 0.3) ^ {0.5} = 0.789 ^{+0.039}_{-0.022}$, while the constraints on the $f(R)$ modified gravity parameters are prior dominated. In the future, the $k$-cut method could be used to constrain a large number of theories of gravity where computational limitations make it infeasible to model the matter power spectrum down to extremely small scales.
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