The end of the drain pipe should be capped, not open.
The bottom of the drain pipe should be perforated. That's where water goes in, not through a wide open end that plugs with dirt.
The drain pipe should be properly sloped, which requires digging up the whole pipe to correct if it's not. About a 1% grade (which 1/8" per foot is close enough to at 1:96) - too flat and water doesn't flow, too steep and water does not carry any dirt that filters in. I assume you'll try not doing that to begin with unless it's a terribly short run, and perhaps it is properly sloped already. If you do dig it out to correct the slope, the drainage pipe should be below the end of the hydrant, not level with it - so somewhat deeper than this.
The whole area of the bottom of the frost-free hydrant should be drainage rock, and that, at least, is usually detailed in the instructors for those. If the soil is silt/clay, the hole should be dug out 12-18" (300-450 mm) around and below the hose supply pipe end, lined with filter fabric and filled with rock, to 12-18 inches above the hose supply pipe end, with the fabric wrapped over the top of the rock, and around the hose supply pipe and drain pipe where they enter/leave that hole.