The word here used for “secluded place” was translated at the top of the chapter as “wilderness”, and in other places as “desolate place”. It includes the implication of loneliness and separation from society, if not simply a hiding place. As mentioned before, the Romans assumed anywhere outside of their domination was “wilderness”, so Jesus starts this chapter in the “wilderness”, and goes to all the towns in Galilee, and ends in the “wilderness”, but it seems he never actually leaves the “wilderness” so long as he remains outside of Roman jurisdiction.
In the Greek, Jesus is not explicitly named. This isn’t actually that weird. In the Greek version, Jesus is only really named in about four verses in the first chapter. The system of pronoun referencing makes it work, and there’s not really any question that Jesus was the intended subject of the sentences. It’s just that his name isn’t actually there.
The activity of Jesus, in Greek, is “προσηύχετο” – prayer to the gods. This is a socially acceptable Roman activity, and probably not remarkable. So why is this detail included? What else is being said? The answer is in the context. It’s the middle of the night, you get up out of bed and go to a secluded place – to void your bowels! There may even be a prayer of relief uttered. What is Jesus doing here in the middle of the night seeking out a secluded space? Pooping?