## hear this ##
"heard that Jesus began to teach in Galilee"
## whether ##
"if"
## the man ##
This refers to Jesus.
## he discovered ##
"Pilate discovered"
## he was under Herod's authority ##
The passage does not state the implied fact that Herod was the ruler of Galilee. You could make this information explicit: "Jesus was under the authority of Herod because Herod ruled over Galilee." (See: [[rc://en/ta/man/translate/figs-explicit]])
## he sent ##
"Pilate sent"
## who himself ##
This refers to Herod.
## in those days ##
This could be translated as "during the Passover Festival."