Hi.
A good authoritative source is The Scots Peerage; by Sir James Balfour Paul, who was Lord Lyon King of Arms for 30 years or so and the authority in Scotland. The specific book for the Cunningham, Earl of Glencairn is vol 4. See:
https://archive.org/stream/scotspeeragefoun04pauluoft#page/234/mode/2up
The mother to Robert was Margaret Hepburn who is shown as the wife to Alexander on Cunningham-2577 but not on his page (Cunningham-4435). Robert must also have been the eldest legitimate son, older than William, as he inherits to Kilmaurs and Glencairn.
Both wikipedia and the DNB carry the same, sourcing one another really.
I have noticed the source for the marriage but the quote does not relate to the same Robert and is not sufficiently extended. The source provides an obligation, in 1425, to marry Anne (Anny) Montgomery to Robert Cunningham, Lord of Kilmaurs (this well before Cunningham-4435 is born. However, on page 87 of that same source (History of Renfrew) it provides an assertion that "Anny" had died and the obligation discharged to Anges, in 1432. In this I notice that Sir James Balfour Paul is incorrect but the real issue is that this relates to the grandfather of the Sir Robert, Cunningham-2060, and not the Sir Robert in question (Cunningham-4435).
It seems that there are a number of duplicates in these families and clearly the incorrect links. If you need a bit of help to sort it out please pass me an email.