In "The Rocking-Horse Winner", does Paul's birthday check solve his mom's problems?

Great question! Clearly the story suggests that it is the
mother's greed and desire for greater material wealth that creates the voices that echo
round the house and haunt Paul so badly - "There must be more money!" However, it is
interesting that when Paul does gain money through his rocking horse and organises to
give it to his mother, she is far from satisfied. Note how she
responds:

He
knew the lawyer's letter. As his mother read it, her face hardened and became more
expressionless. Then a cold, determined look came on her mouth. She hid the letter under
the pile of others, and said not a word about
it.

Later we are told that
she asks the lawyer for the entire sum straight away rather than agreeing to be given it
bit by bit. Then a strange incident is reported about the voices. In spite of the large
sum that his mother has been given, the voices become
louder:

And
yet the voices in the house, beyond the sprays of mimosa and almond blossom, and from
under the piles of iridescent cushions, simply trilled and screamed in a sort of
ecstasy: "There must be more money! Ohhh; there must be more money. Oh, now, nowww!
Nowww - there must be more money! - more than ever! More than
ever!"

Thus it is clear that
in response to this sudden gift, the mother's greed and desire for more is worse than
ever. Lawrence is clearly showing how materialism is a vicious cycle, where when you
gain more you simply want more, which of course in this story results in the death of
Paul as he is driven to ever-greater efforts to find out
winners.