The unclued lights (one of two words and another hyphened) can be arranged to form a word pyramid of 3 to 11 letters, adding one letter and making an anagram step by step.
 

Across

 
11    Demands performances from the past? (6)
13    Totally obsessed having cleared the plate (7, two words)
15    Land where heated bath’s popular (5)
16    Recent cut in coffee (5)
17    Open unusual Scottish rucksack outside (6)
18    Tipsy round rear of shop (5)
20    One who reported home (6)
21    Passageway in centre of Scottish burgh (5)
22    Dog having baked beans — just ideal for starters (7)
27    In addition, different alien is cycling (7)
29    Pigment some brooch reveals (5)
32    Uncle’s first seen in Luigi’s capital, returning for affair (5)
34    Acid spilt right on daughter’s personal documentation (6, two words)
37    Precentor’s unfinished poem (5)
38    Tense pack-animal circling fragrant Japanese tree (7)
39    Acknowledge old edition of Bible. That’s painful! (6)
40    Some weekend supplement (3)
41    Revival of injured necessarian (11)
 

Down

 
1    Heard to verify reserve banking facility (10)
3    Nothing known of new tailless angora in reserve (9)
4    Developing one’s argument about right settling abroad (12)
6    Strap cut – failing to come up with training on this? (8)
7    Poor sister from county cut wine (5)
8    Goose River (4)
9    Northern song about Nationalist’s fictional land (6)
10    SE Asian creature worse if riled (5)
12    Raised rim on table for trifle (6)
14    Bacteria affected two cots. Cripes! (12)
23    Jumper with badge on train (8)
24    Happened to trip. Happened outside (8)
25    President’s upset, having lost king-beetle (6)
26    Type of old black marble for match (7)
28    Leaf-hopper chewed this rose petal for starters (6)
31    Conspicuous success in Quebec, lately (5)
33    Scottish vicarage two leagues away from F1 champion (5)
 
A first prize of £30 for the first correct solution opened on
21 September. 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 2227, The Spectator, 22 Old Queen Street, London SW1H 9HP. Please allow six weeks for prize delivery.