This is the song playing on the stereo when Richard returns home from Stars Hollow. It plays over him looking sad while alone in his study, until the end of the episode.
The opening lyrics are in tune with Richard’s melancholy mood:
I was a stranger in the city
Out of town were the people I knew
I had that feeling of self pity
What to do, what to do, what to do?
The outlook was decidedly blue
A Foggy Day was composed by George Gershwin, with lyrics by Ira Gershwin. It was introduced by Fred Astaire in the 1937 musical comedy film A Damsel in Distress, loosely based on the 1919 novel of the same name by P.G. Wodehouse, and the 1928 stage play written by Wodehouse and Ian Hay. Astaire’s recording was very popular in 1937.
The song has been covered numerous times, and Richard listens to a Frank Sinatra version. Richard may be listening to his 1954 album Songs for Young Lovers, produced by Voyle Gilmore (!), or from the 1961 Ring-a-Ding-Ding! The second one, which was well-reviewed and went to #4 in the charts, seems more likely, as Richard would been about eighteen when it came out.