The Term for Singing in a Very Sad Way

Solution 1:

I don't think we're going to find a verb to specifically mean sing a song of grief (although I hope someone discovers one).

However, there are two nouns - elegy and dirge - which describe such a song. And there is the verb lament which is broader than just singing, but would include singing.

to sing an elegy - I think the best option to describe the form of music in your example

a sad poem or song : a poem or song that expresses sorrow for someone who is dead

http://www.merriam-webster.com/dictionary/elegy

to sing a dirge - could work, but often implies funeral

a song or hymn of grief or lamentation; especially : one intended to accompany funeral or memorial rites

http://www.merriam-webster.com/dictionary/dirge

lament - if you want to describe the way in which someone is singing in your example, then I think this would fit

Express passionate grief about

http://www.merriam-webster.com/dictionary/lament

Solution 2:

Keening (from the Irish) is a possibility.