I came across a web site in which I found that the author wrote this:
To grab a responsible and challenging position in social or education services offering hands-on experience with youngsters.
So, I am not very sure when we should use “offer with”.
(While I saw the Google dictionary saying “offer” should follow by either “to” sb. or “for” sth.)
In the sentence fragment you have posted, the relationship is between the experience and the youngsters. “Hands-on experience with youngsters” is functioning as one unit, and that is what is being offered.
It is like saying “I am offering my talents.” - Fine, right?
“I am offering my talents for writing.” - Now “talents for writing” is a unit just like “experience with youngsters.” “For writing” and “with youngsters” are just telling what the type of talent or experience.
To show you that this is not interfering with the function of “offer,” I am going to combine this with your idea.
I am offering experience with youngsters to Sally. - We use “to” now because we have a specific person in mind as the recipient. As you see, we are offering that unit to Sally. “With youngsters” is only telling us what type of experience is being offered. It does not interfere with the function of “offer.”
“Offer” is used with “to” and “for” when we intend to provide someone something or to make an exchange. “To” is for the former and “for” for the latter.
I will offer my scarf to Alice. - I want to give it to her.
I will offer Bill my favorite baseball card for his exclusive football card. - Exchange
The first sentence’s prepositional relationship is between the scarf and Alice; the second sentence’s is between the baseball card and the football card.
Other uses with “with” in the same sentence:
I offered a $50 donation with Sally (to the Red Cross). - Sally and I offered it together. I offered money (together) with a bike (for his video game console). - I offered both these things.
The parenthetical parts can be omitted. They are just there to help you understand that “offer” is still functioning in the same manner you expect, whether they appear or not.