、。
If I happen to be late, start dinner without me.
Literal
By-chance late-came-[conditional-たら], me without [ぬき-で] dinner [object-を] begin [command-なさい].
ぬき ('without, excluding') is a noun-forming suffix from 抜く ('to remove'), here paired with で to form 'without [someone/something].' 私ぬきで is the set pattern 'without me.' The instructional なさい is typical of parent-to-child or elder-to-younger register.