Diving into the book of John, we look at the continuing ministry of Jesus as he travels back to Galilee.

To review, Jesus had been in Jerusalem, and overturned the money changer tables. While there, He met with Nicodemus to talk about rebirth. Afterward, He and John were in the vicinity to baptize. Jesus began traveling back to Galilee, but went through Samaria, offering the gospel to the woman at the well, and others in town. In doing so, he spent 2 days there before departing.

Today's text begins at John 4:43.
43 After the two days he departed for Galilee.
44 (For Jesus himself had testified that a prophet has no honor in his own hometown.)
Jesus continued on to his destination in Galilee, but he didn't seem to be expecting much from his home town as he approached. Its possible his statement wasn't prophecy, but John included it to foreshadow the rejection that waited for him there.

Spoiler alert. John never records any event that transpired in Galilee as other gospel writers did.

The miracle to be mentioned took place at least a couple days journey from town.
45 So when he came to Galilee, the Galileans welcomed him, having seen all that he had done in Jerusalem at the feast. For they too had gone to the feast.
Apparently, some men from Galilee were in Jerusalem, and witnessed what Jesus had done there. By their welcome, they anticipated the same kind of teaching and miracles. It wouldn't work out that way, but one man couldn't afford to wait.
46 So he came again to Cana in Galilee, where he had made the water wine. And at Capernaum there was an official whose son was ill.
it isn't known why Jesus returned to Cana, the place of his first miracle. Possibly he went to look for more disciples. Possibly it was due to having family ties.

The official is thought to be basilikos, a member of Herod's court. If so, he would be a gentile. This means that in Jesus earliest ministry, he went to Jews, Samaritans, and gentiles. His main mission was to focus on the Jews though.
47 When this man heard that Jesus had come from Judea to Galilee, he went to him and asked him to come down and heal his son, for he was at the point of death.
The official had definitely heard of Jesus healings. The Greek word for "asked" implies insistent, or persistent asking. His request was to come to heal his son..

His son was clearly very sick, at the point of death, and too sick to travel. He could die at any moment. There was no getting better without outside help.

48 So Jesus said to him, "Unless you see signs and wonders you will not believe."
People need evidence, a sign of some kind before they believe. Not that Jesus would hold back, but his ultimate goal is belief. Trust in him.
49 The official said to him, "Sir, come down before my child dies."
The man was desperate. The request to come in person is repeated. The Greek verb tenses make it plain his son is dying. not might die. The son would definitely die if something didn't manifest to interrupt the process.

Did the man need a sign for belief? He trusted enough to know Jesus presence could heal.
50 Jesus said to him, "Go; your son will live." The man believed the word that Jesus spoke to him and went on his way.
Jesus words presented a crossroad of faith, a crisis of belief. A long distance healing was in order to heal the son. Travel time would vibe critical The man had to either believe his son was healed, and go back home... Or abandon faith that Jesus could heal.

He believed, he trusted. He left without Jesus.
51 As he was going down, his servants met him and told him that his son was recovering.
52 So he asked them the hour when he began to get better, and they said to him, "Yesterday at the seventh hour the fever left him."
As he was going down, returning to Galilee, he had good news. His son was living, and getting better. The man had presence of mind to ask the time the recovery began.

BTW.