The Morgoth-element.

[Mithadan]

How much of this element was likely present in the different races

With the exception of the corrupted races, Orcs, trolls, etc., none. Elves, Men (and Hobbits), Ents and Dwarves were not created by Morgoth. They could be "corrupted" by him and his influence, lies and direct attention, but they were not, as a matter of creation, influenced by his "element".

Of course, one can discuss the nature of Arda Marred (which is, at root, a Hebraic concept; there is a specific phrase relating to the act of attempting to cure the ills of the world) that could be deemed to affect all creative (or sub-creative) acts following Morgoth's original corruption of Arda. However, Tolkien emphasizes that the origin of Elves and Men and his adoptive children, Dwarves and Ents (odd that they were devised by spouses; never thought of that) was of his own thought.