The exact wording of Khmer's ability is:
"Holy sites are granted major adjacency with rivers, a culture bomb, food equal to its adjacency, and +2 housing if on a river".
It seems that a more precise wording would be: "Holy sites are granted a major adjacency with rivers, a culture bomb, and food equal to its adjacency. Also, separately, it gets +2 housing if on a river". You don't need to be on a river to get the other stuff, it seems. I wasn't sure, so I tested it. You probably still want to be on a river anyway, but if you have desert folklore or dance of the auroras, I could see some occaisonal areas where you'll get more adjacency without one