In ten clues the wordplay leads to the answer plus one extra letter. These letters in clue order give a name (two words) to which are questionably attributed the unclued lights (in Brewer), which make three phrases: one of six words, one of three, one of four (thematically five?).
 

Across

4    See me in especial trouble, not all at once (9)
10    Confused playwright about to get between the sheets (10)
11    One stranded, pot appearing empty (6)
12    ‘With band in hair’ — poem in neat Scottish (7)
14    From Lincoln, obtain produce (5)
15    Pope is to agitate, forgetting Latin (5)
16    Off course? Just like carrier! (6)
22    Rodent suppresses intention to glide down (8)
24    Finally achieve gentle old age (4)
25    Benevolent cousins died (4)
27    Translated tragedian shortly broadcast (7)
29    Washing boa until restored (8)
32    Lied about one’s knowledge (6)
35    Murderer to be executed, or worse (5)
36    Once sees alien being (7)
37    Travellers’ language in place of refuge spoken (6)
39    Bill visited to sort out some ready-meal ingredients (9)
40    Stock theme best, therefore put back (5)
 

Down

1    By chance mail scrapped without intelligence (10)
2    Screen pulled out of embarrassment (5)
3    Begged in speech to be Shakespeare’s match (6)
4    Imposing building stared at, drunk (7, hyphened)
6    A division of the population number among the best coming up to college (7)
9    Idled away the month of March (4)
13    Be scrupulous and back off fish (7)
15    Dangerous water in river helps to control pressure (6)
17    Bad language, taking beers during short trip (10)
18    Parent waits nervously for artist (7)
19    Super-luxury rating for hotel, The Plough (10, two words)
20    After check, proceeded gingerly, blunt at the front (9, hyphened)
21    Feeble drunkard knocked over in fencing thrusts (6)
26    Tinker managed to cut business into two (7)
28    In legs is a dreadful sensitivity to pain (7)
30    Fruit: rare apple, look (6)
31    Made mixed bottles by halves (5, two words)
34    Was aware of lifting a rail (4)
 
A first prize of £30 for the first correct solution opened on 26 June. There are two runners-up prizes of £20. (UK solvers can choose to receive the latest edition of the Chambers dictionary instead of cash —
ring the word ‘dictionary’.) Entries to: Crossword 2313, The Spectator, 22 Old Queen Street, London SW1H 9HP. Please allow six weeks for prize delivery.