The hope to receive some useful

English as a Second Language English Vocabulary, Grammar and Idioms
1
  1. The hope to receive some useful feedback.
  2. They hope receiving some useful feedback.

What is the difference between these two sentence?

2

Hi, #1 is correct, but #2 is not.

Whether to use a gerund or infinitive as an object can be tricky. These lists maybe be helpful:

englishpage.com/gerunds/gerund_list.htm
englishpage.com/gerunds/infinitive_list.htm

As you can see, the verb hope is on the “uses an infinitive object” list.

An alternative way of using receiving is “They have hope of receiving some useful feedback.” Here receiving is used as the object of the preposition “of”, so the gerund is correct.