This is an English pudding (or cake). A persimmon is a fruit with red-orange skin and flesh which comes in several varities and are in season from October to around Dec or Feb depending on your location. When selecting persimmons, look for plump, soft fruit but not mushy.
Some helpful notes from our Zaar friend Maito, (who so kindly volunteered to be the test kitchen for this recipe that I found scratched down on paper from long ago and had not made personally) are as follows: Either of the two varieties of persimmons will work with this, as long as the fuyus are soft or hachiyas are super soft/almost mushy. Put greased parchment paper (greased side down) under the foil (so aluminum doesn’t drip into your cake). It is also helpful to have a string to make a handle to help you lift the cake out (we did a crisscross around the whole pan). Before step 3, you might measure the amount of water you want first (with the pans and pot in place), before putting the batter in.