A search for the pair production of heavy spin-
1
/
2
or spin-
3
/
2
resonances (
t
*
) in proton-proton collisions at
s
=
13
TeV
is presented. Data collected with the CMS detector at the CERN LHC from 2016 to 2018 corresponding to an integrated luminosity of
138
fb
−
1
are used. The analysis targets benchmark signal scenarios where one
t
*
decays into a top quark (
t
) and a photon (
γ
), and the other into a
t
quark and a gluon (
g
), i.e.,
p
p
→
t
*
t
*
¯
→
t
t
γ
g
. All-hadronic final states from the
t
pair decay chain are selected using jet substructure techniques. The signal is probed as a function of the
t
*
candidate mass, which is reconstructed using the photon and a top quark candidate jet. No significant deviation from the background-only hypothesis is found. Observed (expected) upper limits on the signal cross section at 95% confidence level are set, excluding masses of spin-
1
/
2
t
*
particles below 930 (930) GeV and spin-
3
/
2
t
*
particles below 1330 (1390) GeV. This analysis marks the first search for heavy resonances in the
t
t
¯
γ
g
channel. Exploiting the high-energy photon to reduce the backgrounds, this search achieves sensitivity competitive with
p
p
→
t
*
t
*
¯
→
t
t
¯
g
g
searches for spin-
1
/
2
t
*
despite the small expected
t
*
→
t
γ
branching fraction.
