No, you cannot claim an expense ("deduction") for amounts you did not receive.
The only way you can claim an expense or a deduction is when you pay for the item in question with after-tax dollars.
When you accept less than your normal billing rate for a service, you are simply offering a discount. Discounts are not "deductible" - you never receive them and do not report them as income.
See Are free services or discounts given to clients tax deductible? for additional discussion on this issue.
