  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.
