Cryptic crossword No 4,020

Across

1. Argue with partner about uncovered debt (6)
2. Bottle can discourage swallowing (8)
3. Proof a mouse hides in the public eye? (6)
4. Maverick lecturer on drug, lying about course (4,4)
5. Son follows religious education including interminable hymn and poems (6)
6. Walkers say path to island copse "almost lost" (6,5)
7. Doctor mainly using it without moving (2,4)
8. Racing breaks prevent phlegm after cycling (8)
9. Officer dispatched over back-to-front kit (8)
10. Help by a revolutionary causes lumbago! (8)
11. Light breeze bearing against face of obelisk (6)
12. Copyright ruined true opening (8)
13. Extracted oil from sample, initially concealed by unknown journalist (6)

Down

1. Daughter given unfinished envelope needing name to make progress (7)
2. Ego gutted as VAR judge stirred up City (8)
3. Drawing aid copies from a novel without permission (4,2,9)
4. Explained how to make froth! (3,5)
5. Pocket watch, essentially one that wakes (7)
6. Law covers question of the French crossing English Channel (8)
7. Figures old seabird's going south (8)
8. Bone's regularly skinned by speed-somersaulting (7)
9. Thought about securing code upon input/output (7)
10. New students at British school in topless picket (6)
11. Per dictator, crown is unfairly decided? (6)
12. "Nearly new" and excluding excessive returns (4,2)
13. Short and fat piano occupies musty shed (6)