I couldn't find any reference to this at all. I am guessing though that it is "old English" and probably arises from common everyday usage.
My wife is a bit of an expert in this field and tells me her idea..
In the days when outer coats were made from heavy woolen material and a regular brushing was required to keep them looking clean, the "Specks"
were dust, dirt, lint or other contaminant that was easily brushed off with a coarse brush. The "Spicks"
were less easily removed being embedded into the material and requiring a more intensive approach to get rid of them.
Speck is still around today e.g. a speck of dust
while I have never heard the word spick
used other than in a racist and derogatory way from the early part of last century in reference to persons of Latin descent. This is confirmed as the "meaning" of that word in many dictionaries.
So I guess the Spicks and Specks
of the song, is a reference to the clutter of everyday life that can attach itself to us and while some is easy to remove others are not so readily dealt with.
What do you think?