Partial answer
The cryptic clues ...
... have no definition. Instead, the title of the crossword puzzle, "Capital Punishment", gives a hint: All answers are national or state capitals.
Across
3 No, that is, in lieu of two, a single beheaded homicide leader precedes
HONOLULU, capital of Hawaii, USA
H(omicide) + (m)ONO + L(ie)U without IE twice, not sure about the order
4 Limits of source removes, intrinsically the energy would be lost
PERTH, capital of West Australia
PER SE - S(ourc)E + TH(e) - E (energy)
5 Tip for buying muppet – by Peter
BELMOPAN, capital of Belize
B(uying) + ELMO + (Peter) PAN
6 A broken shank is leading
KINSHASA, capital of the Democratic Republic of Congo
SHANK IS* + A
8 Demolish Cal's cabana
CASABLANCA, principal city of Morocco
CAL'S CABANA*
10 Call me after one
DUBAI, capital of the Emirate of Dubai in the UAE (Thanks to Alconja!)
DUB (call) + A + I
Down
1 Sly glares I received
ALGIERS, capital of Algeria (Thanks to Weather Vane!)
GLARES* around I
2 False hope with early sign of new Chinese leader from the east
PHOENIX, capital of Arizona
HOPE* + N(ew) + (Jinping) XI<
3 Relish scattered moment. Embraced? Behold!
HERMOSILLO, capital of Sonora, Mexico
RELISH* around MO + LO
7 Member of travel amateurs returned
MALE, capital of the Maldives
trav_EL AM<_ateurs
9 A mind, having taken in ecstasy
SEOUL, capital of South Korea
E (ecstasy, drug) in SOUL
The grid:
A
P H O N O L U L U
P E R T H E G
O R I
B E L M O P A N E
N O R
K I N S H A S A S
X I
L M
C A S A B L A N C A
E O L
O E
D U B A I
L
The overall answer:
There is an answer of eleven letters (two words: 5, 6) to this crossword puzzle. I haven't found it. There are eleven clues whose first letters are APHMS HPBKCT and eleven common lights: HLHEM ISSLAU. Using the states that the answers are capitals of doesn't yield anything promising, either.
It's also not clear whether the answer is another capital of whether it fits with the Halloween theme. The answer could also be a punishment, as hinted at in the title. Kuala Lumpur and Addis Abbaba are capitals that match the letter count (5, 6). Buenos Aires is a near miss. I don't see how to make any of those, though.
For what it's worth, here's a list of the answers:
Honolulu Hawaii US-HI
Perth Western Australia AU-WA
Belmopan Belize BZ
Kinshasa Congo CG
Casablanca Morocco MA
Dubai Dubai AE-DU
Algiers Algeria DZ
Phoenix Arizona US-AZ
Hermosillo Sonora MX-SON
Male Maldives MV
Seoul South Korea KR