hope to

  1. I hope you lead him to pay more attention to sales work.
  2. I hope you to lead him to pay more attention to sales work.
  3. I hope that you will lead him to pay more attention to sales work.
  4. She is hoping to win the gold medal.
    Please correct them if they are wrong.

#2 is wrong.

#4 is OK.

#1 and #3 are grammatically correct, but this use of “lead” feels a bit unusual. More usual would be something like “I hope you encourage/persuade him to pay more attention to sales work”.

#2 was the original, taken from a letter which you read on this forum, wasn’t it Allfathima.

Do you think Luschen and I would both have changed the original if it wasn’t wrong?
Do you think Luschen would have suggested #1 if it was wrong?

I agree with Dozy about #1. As Beeesneees said, this was from a letter I gave a suggestion about and I tried to keep the phrasing as close as possible to the original letter.