Hola Pati

cualquier/a and algún/a are very similar but not always interchangeable. 

Cualquier has that specific nuance of "it doesn't matter which or what", as we define it here in this lesson. The problem is that I don't think the English makes that different very obvious, and uses "any" for both. 

If I say:

Puedes coger cualquier pastel. 

What I mean here is that you can pick "any" as in whichever you like, no limitation, you choose. 

But if I say:

Puedes coger algún pastel.

Although the options are unlimited, "algún/a" doesn´t carry that nuance that cualquier has. 

In the sentence from the writing challenge: " I'll see if I have any chance with him." we most likely go for "alguna" or "una" because that sentence is not really offering that interpretation that I mentioned before. It's just saying "a chance,  some chance". 

Saludos