It actually is not a hose but a wire loom which appears to connect to the starter. It may be easier to see in the first picture below.

The 2nd picture is of the same heat shield just below the outer rear of the head looking in from behind the front tire of the passenger side. I cleaned it all up last evening, just had to merely wipe it with a rag. After driving about 50 miles today this is what it looks like. Appears most of the oil just slides off of it onto the belly pan/reinforcement plate.

Attached Images