What does ‘crack on’ phrase means. I've heared it in following use cases.
- Boss is saying to us: 'Let's crack on' - and we start discussing projects.
- Man is discussing with sombody: 'We've cracked on with that girl'
Can somebody explain it please.
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It appears to be British slang (at the very least). It also appears to have several meanings, but the relevant ones follow.
For the first one
Let's crack on
So it sounds like it could mean, "Let's begin!" or "Let's continue!" depending one where you are in the process.
We've cracked on with that girl
- to try to become familiar with someone you would like to date. 
So it sounds like they hit on/flirted with the girl.
to get going
and is indeed a very BrE idiom, The AmE equivalent would be
Let's get cracking!
However I think if one said
We've cracked on with that girl.
I've cracked on with that girl.
it would be understood to be much more than just flirting and would have a different meaning entirely. A bawdy barman I knew in London used to say jokingly to the Polish waitresses
Do you want to go to Kraków (pronounced 'crack-off') with me?
one of the waitresses, who was more in the know, once replied
I think you can do fine all by yourself!
he was using it as a double entendre.
There are several uses for the slang term ‘crack’ in the UK vernacular and it really depends on the context.
‘Let’s crack on with...’ is accepted to mean ‘Make progress’ either by starting or resuming the activity that follows the phrase... eg - let’s crack on with the report means let’s get the report started, or let’s get moving on the report.
‘He’s onto a good crack’... here the word is used as a noun. The crack substituted for the word ‘thing’ ... it might be a car deal or other pursuit, but it usually tends to refer to employment of some sort and ‘a good crack’ might either be a well paid job or one with particularly good benefits.
Crack - Commonly used in reference to cocaine.
I can see your ‘crack’ in those jeans. A noun referring to the channel between the two muscles (gluteus maximum) that make up the bottom/behind/rear/bum.
You ‘crack me up’ generally means I find you very funny and you make me laugh - a lot.
‘Crack on with’ ...a girl (or boy) - is similar to point 1 and it means ‘start with’ but in more specific terms of romantic or sexual intention, to start chatting up another individual with further intention.
Boss is saying to us: 'Let's crack on' - and we start discussing projects.
This means continue and is usually used when time is being wasted or focus is not on the important task. So you might be discussing a new system at work and the conversation drifts to an old system and boss gets it back on track with 'let's crack on' i.e. forget about all that crap we have more important things to do.
Man is discussing with somebody: 'We've cracked on with that girl'
Would need to know the context but could be a similar usage to the boss scenario.