"My requesting" = the act of my making a request "My request" = the request which I make "I hope my requesting doesn't offend you" = I hope that you are not offended that I would request something from you "I hope my request doesn't offend you" = I hope that you are not offended by what I asked for

Requesting is a verb (an word that is an action). It is something you can do. Request can be a verb or a noun (a noun is a word that is an object or idea) In your example: My request was turned down. is the the grammatically correct way to say it. The -ing at the end of a word means that it is being done as the word is spoken. For example, I am typing right now. You were requesting an answer

If there's something that has been asked for, it's a request, and considered a noun. "Did you get my request for those days off?" Request is a noun here. In your example, "My request was turned down," is correct. Requesting is being in the process of making a request. Requesting is an action, which would be a verb tense. "They are requesting more time to approve the request I made the other day." Requesting is the verb and the request is a noun.

The suffix -ing, indicates that the word is a verb. So unless the sentence you were doing the action of requesting something, the word request is better to use, as it is a noun, not a verb.

Request has both a noun and a verb form, in the sentence above it would be "request" and would be in the noun form. "Requesting" is a verb and would not fit in the above sentence.