Yes - if you are reporting your winnings as self-employment income, then you will be able to claim any "ordinary" and "necessary" expenses related to making this money. Your expenses can include feed, hay, travel expenses, entry fees, etc. (If reported as hobby income then you will only be able to claim expense up to the amount of hobby income being reported.)
Please
refer to this IRS link for more information about Business
Expenses
To
enter Business income and expense information in TurboTax, log into your tax
return (for TurboTax Online sign-in, click Here and click on "Take me to my
return") type "Schedule c"
in the search bar then select "jump to
Schedule c". TurboTax will guide you in entering this information.