Differential cross section measurements of
W
±
γ
production in proton-proton collisions at
√
s
=
13
 
 
TeV
are presented. The data set used in this study was collected with the CMS detector at the CERN LHC in 2016–2018 with an integrated luminosity of
138
 
 
fb
−
1
. Candidate events containing an electron or muon, a photon, and missing transverse momentum are selected. The measurements are compared with standard model predictions computed at next-to-leading and next-to-next-to-leading orders in perturbative quantum chromodynamics. Constraints on the presence of TeV-scale new physics affecting the
W
W
γ
vertex are determined within an effective field theory framework, focusing on the
O
3
W
operator. A simultaneous measurement of the photon transverse momentum and the azimuthal angle of the charged lepton in a special reference frame is performed. This two-dimensional approach provides up to a factor of ten more sensitivity to the interference between the standard model and the
O
3
W
contribution than using the transverse momentum alone.