There is an aeroplane carrying a British family, heirs to a vast fortune, flying to South Africa and everything seems to be going well until they hit turbulence over one of the plains. In fact, so badly do things go that the craft crashlands into the jungle and kills everyone onboard - everyone but the couple's infant son. The local chimpanzees take an interest in the wreckage and scoop up the baby, whisking him into the trees while a tribesman investigates the inside of the plane. But Cheeta, chimp friend of Tarzan (Johnny Weissmuller), takes the baby away from his fellows and back to the Apeman's treehouse...
You'll note that this film was called Tarzan Finds a Son! and not Jane Gives Birth to a Son! because the censors would not allow an unmarried couple in the movies to have children, this in spite of how the famed couple couldn't keep their hands off each other in the previous three films. So the solution was not to have a clergyman parachuted into the jungle (although Jane refers to Tarzan as her husband at one point), but to have them adopt instead, hence the appearance of little Johnny Sheffield as the imaginatively-named Boy.
Once it has been established that Boy has settled in very well with his new "parents", there has to be a threat to their idyll, and that happens to be the Lancings, the true family of the child. They are out to track down the crash site and take back the survivors, if there are any, and once they encounter Tarzan and company they are set to wondering about the boy. Jane may assure them that he's her son, but they remain unconvinced, and none too pleased at the reception they have been treated to by the unaccommodating Lord of the Jungle.
"White people!" spits Tarzan when he sees a bullet wound in his favourite elephant's side, and it's true that whenever they turn up it spells more grief for him and his loved ones. This means yet more smashing up of guns, his pet hate, and he even goes to the lengths of stealing and dropping the rifles into a hard-to-get-to pool (the Brits have photographic evidence it was him, as if any were needed). But it's Jane who proves the biggest letdown to our hero when she realises Boy should be living with his real family and not living it up with baby elephants.
Tarzan's relationship with Boy is interesting in that he is supposed to be a father figure to him, but in Cyril Hume's screenplay he treats him on essentially the same level as Cheeta, as a pal to play with rather than someone to act as guardian to. As is well known, this was because Boy was being set up as a companion substitute for Jane because of Maureen O'Sullivan's increasing reluctance to return to the role. In fact, Jane was supposed to be killed off at the end of this one and either fan pressure or the Edgar Rice Burroughs estate put a stop to it. What this leads up to is an ending that obviously has to have Jane die, only for her to pull through at literally the last minute, which is offputting whether you wanted her gone or not. More money for O'Sullivan secured her services for two more instalments and while this is entertaining enough, you could tell they were already running out of ideas.