v1.0
Publishing history:
v1.0: 20/07/25
cudaig m. 
The gender of variant forms cited below is as follows: cudaig m., cudaige m., cudag f.; cudainn f., cudan m.
MacBain (1911) associates SG cudaig with Ir cudóg, códog [leg. codóg] ‘haddock’ (cf. Dinneen 1904), for which he suggests the etymology *cod-do-. However, cudóg and codóg are variants of Ir. cadóg, the normal word for ‘haddock’ in Munster and Connacht (Ó Baoill 1994, 181), contrast hadóg in Ulster and feadóg in Donegal (ibid., 182), all from Eng. haddock.
So SG adag.
SG cudaig refers to ‘a cuddy, i.e. a young saithe or coalfish, Pollachius virens’, and has a number of variant forms:
A. SG cudaig, cudaige, cudag
A 1. SG cudaig
In HSS 1828; MacEachen 1842, s.v. cudainn; Forbes 1905, 41, 354; Mackenzie 1910, 384; MacBain 1911; Dwelly 1911: see cudainn; Faclan bhon t-Sluagh˄: Lewis, Skye, Sutherland, Ross-shire: cudaig-ruadh; AFB˄.
A 2. SG cudaige
In Forbes 1905, 41, 354; MacLennan 1925: s.v. cudag; Wentworth 2003: [kʰutik’ə], Gairloch; Faclan bhon t-Sluagh˄, s.v. an cudaige, an saoidhean, an t-umhcas: Applecross, and s.v. cudaice: Scalpay.
A 3. SG cudag
In Dwelly 1911: see cudainn; MacLennan 1925. From a non-native Gaelic speaker fascinated by the Gaelic influence in south Ayrshire, Faclan bhon t-Sluagh˄ (s.v. cudag) slips ‘Other [Gaelic-originated words that come to mind] are scart = scarbh = parkin = partan = pickie = young saithe, also piteuchty (Girvan). This fish is the cudag – cudaige – cudann at the sprat stage and living in the harbour. ...’ [sic], although it is not always easy to differentiate Gaelic from Scots forms.
B. SG cudainn, cudan
B 1. SG cudainn
(i) cudainn (Shaw 1780: cudinn; MacFarlane 1817; HSS 1828: see cudaig; McAlpine 1832; MacEachen 1842; Forbes 1905, 41, 354; Dwelly 1911; MacLennan 1925; Dieckhoff 1928: [kudiNˈj], Glengarry; Faclan bhon t-Sluagh˄: Tiree, Coll, Islay, North Argyll: [ku̜d̪ĩɲ]; AFB˄); 
Farran (1946, 12) cites the Scots form cudden for SG cudainn.
(ii) cuideann (Faclan bhon t-Sluagh˄, s.v. piocach, saoidhean, saoidhean mor = Sherlock: Mull);
(iii) cudann (from a non-native Gaelic speaker fascinated by the Gaelic influence in south Ayrshire, Faclan bhon t-Sluagh˄ (s.v. cudag) slips ‘Other [Gaelic-originated words that come to mind] are scart = scarbh = parkin = partan = pickie = young saithe, also piteuchty (Girvan). This fish is the cudag – cudaige – cudann at the sprat stage and living in the harbour. ...’ [sic], although it is not always easy to differentiate Gaelic from Scots forms).
B 2. SG cudan
(i) cudan (Armstrong 1825; Forbes 1905, 41; Dwelly 1911: see cudainn; AFB˄);
(ii) cudain (Faclan bhon t-Sluagh˄: Mull: [kutɑ̣˖nʹ]).
SND˄ compares Scots cuddie and cuddin with, while OED˄ derives ‘chiefly Scottish’ cuddie (3cuddy 2.) and ‘local’ cuddin (cudden 2a.) from, SG cudaig and cudainn. However, the direction of borrowing is more likely to have been the other way: ON *kóð nt. yields Scots kød (SND˄; Jakobsen 1928) and, with /d/ also interchanging with /đ/ between vowels, 
See Macafee and Aitken˄ §6.31.4.
MScots hirling, perhaps an adaptation of Eng. herle ‘the barb of an angling fly’ (SND˄).
I.e. within Scots itself. For discussion of final -in ~ -ie alternation in Scottish Gaelic originated place-names, particularly in the east of Scotland, see Ó Maolalaigh 1998, 30–38: on the question of the origin of the Scots diminutive suffix -ie from p. 34, on the Early Gaelic diminutive suffix -ín from p. 36.
Scots cuddin yields SG cudainn (B 1) and, with an alternative (diminutive) suffix, cudan, locally cudain (B 2). The singular form cuideann (B 1(ii)), with transference of palatalisation, appears to be in error or idiolectal for cudainn; cudann (B 1(iii)), if it is meant as a Gaelic form, is probably for cudainn, either as a back-formation or in error.
Scots cuddin (cudding), or perhaps in some cases SG cudainn itself, yields SG cudaig, cf. SG feòirlinn ~ feòirlig ‘farthingland’, bòrlainn ~ bòrlaig ‘a strip of arable land’, farsainn ~ farsaig ‘wide’ (s.v. feòirling), occasionally cudaige, 
?By analogy with the ending -ige, found in a few loan-words, e.g. SG mastaige ‘an unpleasant person’ (Faclan bhon t-Sluagh˄; ?< Scots masta(d)ge ‘mastiff; also applied to a person as a term of abuse’, variant of mastis (SND˄)) and rustaige ‘an unpleasant old man’ (Faclan bhon t-Sluagh˄; ?< Scots or Eng. rustic, used substantivally), and compare the substitution in cnèibige for cnèibilt ‘garter’ (s.v.).
SG cudainn has a more southerly distribution, while cudaig has a more northerly one. Ir. cudainn and (Antrim) cudán 
But Holmer 1942, 181: (Rathlin) [kʎdən].
So Farren 1946, 12: cudainn, cudan.
Derivatives: SG cudaigeachd ‘rock-fishing for cuddies’ (Faclan bhon t-Sluagh˄: Lewis; AFB˄), a variant of *cudaigeach, cf. iasgachd for SG iasgach ‘fishing’.