I have searched the internet extensively to find an answer to no avail. These notes cannot possibly be played at the same time so my theory would be the note can be played as a quarter beat or a half beat but then, why would it even be put in there in that form, surely it would be one or the other, end of. Any one who can shed light on this would greatly ease my curiosity.
Of course you can't play a unison on piano. This note belongs to two voices which happen to come in unison. This is how voices are notated in polyphonic music.
In your example, in the second measure of the second system the right hand plays:
- F on the first beat
- holds F on the second beat while playing E
- D-F on the third beat
- on the fourth beat holds D while playing G.
If the score is a transcription, then in the original perhaps the voices were played by separate instruments, which could play the same note in unison. On piano you can play it only as a single note, but knowing it belongs to two voices may help you to articulate it.
The reason for that is that there are two different voices and the music is illustrating each part independently, melody and harmony. If you wanted to assign these parts to a guitar and bass or a sax and trumpet you would need this information.
On an instrument like the piano you cannot play two copies of the same note like this so you just play it once. On the guitar it is common for unisons to be notes and played as doubled notes. This is easily done when the note corresponds to an open string, then both can be played at once.