As a small business can I write or deduct anything if I sponsor a local athlete? Examples Reg Fees, Clothing? So long as they were advertising for my business?

Yes, you would deduct this as an advertising expense.  

You may deduct expenses for advertising your business. It's important to note that these must be ordinary and reasonable expenses for advertising.

https://www.thebalance.com/what-business-advertising-expenses-are-deductible-398945

Below is from IRS web site:  https://www.irs.gov/publications/p535/ch11.html

Advertising expenses.   You generally can deduct reasonable advertising expenses that are directly related to your business activities. 

  You can usually deduct as a business expense the cost of institutional or goodwill advertising to keep your name before the public if it relates to business you reasonably expect to gain in the future. For example, the cost of advertising that encourages people to contribute to the Red Cross, to buy U.S. Savings Bonds, or to participate in similar causes is usually deductible.

So in the situation above, would the company pay for things like Registration Fees and other expenses that the athlete incurs? Or would the company write a check to the athlete directly? 
What the athlete need to count that check as personal income? 

@neonprairie - I am not an expert in this area, however, since the NCAA permitted student athletes to sell their Names, Image and Likeness (NILs), this area has caught the IRS's attention.  See link below.

 

in the post above, the OP asks if he can pay the athlete's expenses in return for advertizing the business on the student athlete's attire.  Sounds like a trade for the student's NIL. 

 

the very first line of this link states: 

 

"Any income from NIL activities, including non-cash, is considered taxable income."  So it would appear to me that any cash  or non-cash items the athlete receives for registration, etc would be taxable income because the athlete agreed to compensation for his NIL.