Kleopatra ist the default certificate administration user interface of Gpg4win. Instead, the help files only mention the GNU Privacy Assistant which is an alternative program for managing certificates, in addition to Kleopatra.
https://gpg4win.org/doc/en/gpg4win-compendium_6.html
The screenshot shows how to enable the putty support.
Meanwhile I work without backup of the private key. This means that, for each server, one needs a method to replace the pubkey in case of loss or damange of the NitroKey.