  Different orientations of $\alpha$-clustered carbon nuclei colliding with
heavy ions can result in a large variation in the value of anisotropic flow.
Thus, photon flow observables from clustered ${\rm^{12}C}$ and ${\rm^{197}Au}$
collisions could be a potential probe to study the `direct photon puzzle'. We
calculate the transverse momentum spectra and anisotropic flow coefficients
($v_n$) of thermal photons from collisions of triangular $\alpha$-clustered
carbon and gold at $\sqrt{s_{\rm NN}}=200$ GeV at RHIC using a hydrodynamic
model framework and compare the results with those obtained from unclustered
carbon and gold collisions. The slope of the thermal photon spectra is found to
vary moderately for different orientations of collisions. However, we find that
the elliptic ($v_2$) and triangular flow ($v_3$) coefficients of direct photons
for specific configurations are significantly larger and predominantly formed
by the QGP radiation. A strong anti-correlation between initial spatial
ellipticity and triangularity is observed in an event-by-event framework of
$\alpha$-clustered ${\rm C+Au}$ collisions. These special features provide us
an opportunity to detect the exotic nature of cluster structure inside carbon
nucleus using the photon probe in the future experiments.
