A mixture of things that offer
A mixture of things that offers
It is not obvious to me which one is correct because mixture is singular but things is plural.
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If the subject is singular, use "offers", for example:
He offers a cleaning service.
If the subject is plural, use "offer", for example:
They offer a cleaning service.
In your example of "a mixture of things", although there are plural "things" they are being collectively described as "a mixture of things". This should be treated as a collective noun. As collective nouns are singular you should therefore use "offers".