Rep. Blake Farenthold, a Texas Republican, used a taxpayer fund to settle a sexual-harassment claim by an employee three years ago, court documents and congressional data show.

The settlement stems from a court complaint by Lauren Greene, the lawmaker’s former communications director, who sued Mr. Farenthold’s office in December 2014 alleging wrongful termination. She said in court documents that she had been fired after complaining to another staff member about what she said was inappropriate behavior and comments by the...