Incorrect: They were counting on me helping them.
Correct: They were counting on my helping them.

Incorrect: They insisted on me resigning the post.
Correct: They insisted on my resigning the post.

Gerunds should be used with possessive pronouns (my, your, her), and not object pronouns (me, him etc). Note that this rule is not strictly observed especially in the case of proper nouns.

Incorrect: I am thinking to write my autobiography.
Correct: I am thinking of writing my autobiography.

Incorrect: The clever girl was confident to win the first medal.
Correct: The clever girl was confident of winning the first medal.

Incorrect: I am hopeful to secure a loan to build my house.
Correct: I am hopeful of securing a loan to build my house.

Certain verbs and adjectives are  followed by a preposition + gerund. Examples are: think, desirous, despair, fond, confident, prevent, abstain, refrain, prohibit etc.

Incorrect: To die with honor is better than living with dishonor.
Correct: To die with honor is better than to live with dishonor.

Incorrect: Alice likes singing, dancing and to paint.
Correct: Alice likes singing, dancing and painting.

Your writing will be clearer if you use parallel structures in both clauses.

Incorrect: There is no hope of the fog’s lifting.
Correct: There is no hope of the fog lifting.

The possessive case should not be used with the gerund when the noun denotes a lifeless thing.