This name is also found as MacDonough and MacDonogh. It is derived from the Gaelic Irish Mac Donnchadha. In Connacht there is unlikely to be any O’Donoghue connection other than by mistake, but this is not the case in Munster.
Seventeenth English records very often confuse the two names and many Ó Donnchadhas were recorded as Mac Donnchadhas. There was a sept called MacDonagh MacCarthy in County Cork who, in my opinion, were O’Donoghues who decided to align themselves opportunistically with the MacCarthys (see journal article in January 2003). It is unusual to find people with this name in Cork these days as most have dropped the MacDonagh prefix.