I would like to know what しっかり頑張ってこい means and if it has any difference (any
extra nuance) with がんばって.

Also:

Why is こい (command form of くる) used?

In which situation do you use this expression?
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question
verbs subsidiary-verbs
Meaning of しっかり頑張ってこい
There was a question just a couple days ago about 呼んでくる, and the meaning of くる
in that case, which might also apply here.

[A little clarification about
呼んでくる](https://japanese.stackexchange.com/questions/63020/a-little-
clarification-about-%e5%91%bc%e3%82%93%e3%81%a7%e3%81%8f%e3%82%8b/63021#63021)

Without knowing the context of the しっかり頑張ってこい it's hard to know for sure
whether the **こい** is as you mentioned, an actual "command to come (back)" or
whether it is くる being used in its "emotional" sense, representing a
psychological "approach" to the speaker, or "growth/change".

The major semantic difference is that **頑張って** (ください) is a simpler well
wishing phrase that less emphatic, and doesn't make assumptions about the
results of the other person's efforts.

**頑張ってこい** on the other hand, by including the "command" form of くる, can be
thought of as more emphatic/assertive, more positive, possibly even predicting
or insisting on the success of the other person.
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It is fine to say がんばって to anyone, but you can only use 頑張ってこい if you are
higher in the honorific status (otherwise it will be rude). This is because
頑張ってこい is a command.

しっかり means "properly", and thus you should also only use it if you are higher
on the honorific status. Both しっかり頑張って and しっかり頑張ってこい is possible. The former
form is considered "softer". For example, women will tend to avoid しっかり頑張ってこい
and instead prefer しっかり頑張って.

〜てこい is a command to perform something. For example, one can also say
洗ってこい、食べてこい、やってこい etc. The difference with 洗え、食べろ、やれ is that it implies you go
somewhere to perform these things, rather than doing it here.

To the point of @ericfromabeno, I agree. If somebody _can_ say 頑張ってこい (due to
the honorific relationship), but chooses 頑張って, then they are expressing the
wish for you to succeed. They won't use this form if they are insisting on
your success. For example,

```

    「三等兵、危険な任務だが頑張って」　←　odd
    「三等兵、危険な任務だが頑張ってこい」　←　natural
    「決勝戦絶対に負けるなよ、頑張って」　←　odd
    「決勝戦絶対に負けるなよ、しっかり頑張って」　←　less odd
    「勝ち負け気にせず頑張って」　←　natural
    
```

I think the main difference is pressure. If you don't want to give someone
pressure, you use 頑張って. If you want to create a positive pressure you add しっかり
or even ってこい. You can only create pressure without being rude if you are
"higher" on the honorific status though.
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answer