  A sample of $1.3 \times 10^5$ $K^+ \to \pi^0 e^+ \nu \gamma$ candidates with
less than 1% background was collected by the NA62 experiment at the CERN SPS in
2017-2018. Branching fraction measurements are obtained at percent relative
precision in three restricted kinematic regions, improving on existing results
by a factor larger than two. An asymmetry, possibly related to T-violation, is
investigated with no evidence observed within the achieved precision.
