In "Through the Tunnel", how has the relationship changed between Jerry and his mother by the end of the story?

I think you would benefit from looking at the story in
much wider terms - certainly the Jerry at the end of the story is very different from
the Jerry at the beginning, and you are right to identify that one way he has changed is
in terms of his relationship with his mother. However, more generally, by the end of the
story Jerry has gone through a journey from childhood to manhood, symbolised most
stridently in his journey through the tunnel.

At the
beginning of the story we are introduced to a character who is on the cusp of
adolescense, and very clearly feels responsible for his mother due to their enforced
intimacy. Yet despite his feelings of responsibility towards his mother, he nonetheless
feels drawn to the "wild beach", which is away from the "safe beach" and his mother's
attentive care. The wild beach here can be said to symbolise independence and life away
from the protection of a parent figure - note how Lessing describes the two beaches to
draw out this comparison.

His discovery of the tunnel and
the challenge that the French boys set him through swimming through the tunnel spur
Jerry on to train hard and eventually succeed in his attempt to go through the tunnel.
Although certainly at the beginning of the story it is Jerry's need to be accepted by
the older group of French boys that drives his desire to go through the tunnel, it is
interesting that at the end of the story he no longer feels this is the case, as he is
happy to go back home and spend time with his mother. This indicates that the tunnel was
more about a process of self-acceptance and doing something to show he could do it for
himself rather than for any other reason.

His relationship
with his mother likewise has changed by the end of the story. Jerry deliberately
witholds his triumph, only relating his ability to hold his breath. The dramatic irony
in his mother's response ("I wouldn't overdo it, dear") indicates the independence that
Jerry has achieved in his journey through the tunnel - he has now entered an arena where
he has secrets from his mother and is able to engage in activities, dangerous activies,
away from his mother's protection. He is no longer subject to her and has proved himself
a man in his own right.