Songs refuse to sync to iPhone SE though they display perfectly in the iTunes desktop app...

1. Do the songs play in your iTunes Library?
2. Are the songs selected for addition to your devices? Are there ticks (checkmarks) in the tick box to the left of the troublesome song titles?
   1. in the Songs view, there is a tick box to the left of each song title. The tick (checkmark) needs to be in the box. You can see all the tick boxes, for each song, in the Songs view:
   2. in other views (such as the Album view) you need to right-click a song and confirm that the menu option includes Untick Selection (or possibly Uncheck Selection). This means that the tick is on. If the menu option states Tick Selection, you need to select it in order to turn the tick on. You will also notice that an unticked song is greyed-out:

If you need to change the status of any song, you then need to Sync your devices to your library for those changes to take effect.