Irregular rhyme usually, as @deadrat mentioned, means something which does not fit into any repetitive rhyming sequence: so a Shakesperian sonnet with line endings AAB CCB DDE FFE is regular, AAB CBC CAB BCA is not. The difference is kinda sketchy, though: is a limerick (AABBA) regular?
So in real terms, regular rhyme just means "fitting with the rule of an established rhyme pattern". If a listener can anticipate the ending of the next line, then it's regular. A good example of a subversion of this for use in doggerel is "The Assumption Song".
AN irregular rhyme, though, can sometimes mean a poem (aka 'rhyme') with perfectly regular rhyme, but with irregular meter. Once again, regular just means "what people expect". And meter is just which syllables or sets of syllables are unstressesd, and which aren't. So a limerick goes something like:
da DA da DA da DA da
da DA da DA da DA da
da DA da DA
da DA da DA
da DA da DA da DA da
(where the unstressed 'da's can be multiple syllables and sometimes even left off)
This fits:
A wonderful bird is the pelican,
His bill will hold more than his belican,
He can take in his beak
Enough food for a week
But I'm damned if I see how the helican!
This... doesn't:
There was a young man from Japan
Whose limericks never would scan.
When asked why that was,
He replied "It's because
I always try to cram as many words into the last line as I possibly can."
[Following included because I misread the question initially as being what Doggerel is]:
Doggerel is often used by poets to describe poetry which doesn't follow the traditional rules of meter and rhyme: which, in short, is what irregular rhyme is.
In layman's conversational English, doggerel just means any poetry - especially naive or earthy poetry - which is not intended to be judged for its artistry or high-minded ideas.
Casually-written limericks, for example, while fitting perfectly to the limerick style, often get called doggerel if their main point is bawdy humor rather than artistry. The same with children's school-yard chants, nursery rhymes, etc.
It's also sometimes used self-effacingly by poets to describe their own work, in the same way as calligraphers might refer to their own work as "chicken-scratch".
Best Answer
I admire your will to contrive history to turn Old Mother Hubbard into a master work of literature. Unfortunately in this case, I don't think there's terribly much evidence for these words rhyming (or at least, not for a majority of speakers). If you look at dictionaries from the 19th century, the words "bone" and "none" are squarely transcribed with different vowels, as are "coffin" and "laughing". (There's a chance that, once upon a time, more speakers did pronounce the vowel of "laugh", and indeed other words such as "dance", similar to that of "coffin", but sadly not at the time Old Mother Hubbard was written.)
If you want to be euphemistically kind to Old Mother Hubbard, then you could call it a "visual rhyme". An alternative theory is that Old Mother Hubbard is actually not a master work of literature. Shock horror.