I am filing separately from my wife this year. She already claimed the house we purchased in April 2022 when she filed her taxes. Do I have to report the house?

No.  You are not required to report any deductions on your tax return.  Having said that, if your spouse is itemizing her separate return, then you must itemize on your return even if your deductions are below your standard deduction (12,950 for married filing separate).  

If you have to itemize you cannot include deductions on your return that were used on your spouse's return (double count).  For example, the total deduction for mortgage interest paid and the real estate taxes paid between your return, your spouses return and the other co-signor cannot exceed the total amount actually paid.  

Verify the deductions on her return form 1040, line 12, and if her return included a schedule A showing the itemized deductions on that line 12.  

If she used the standard deduction then you must do the same.  Typically, married filing separate returns do not have enough itemized deductions to benefit both filers and therefore the standard deduction is used.  

If indeed she itemized, then you need to enter your portion of itemized deductions (medical out-of-pocket, state/local taxes paid, charitable donations) and choose itemized deductions even though TurboTax may say "Standard is better."