We theoretically and experimentally investigate high-harmonic generation in a 78-MHz enhancement cavity with a transverse mode having on-axis intensity maxima at the focus and minima at an opening in the following mirror. We find that the conversion efficiency is comparable to that achievable with a Gaussian mode, whereas the output coupling efficiency can be significantly improved over any other demonstrated technique. This approach offers additional power scaling advantages and additional degrees of freedom in shaping the harmonic emission, paving the way to high-power extreme-ultraviolet frequency combs and the generation of multi-MHz repetition-rate-isolated attosecond pulses.