When do we translate 'den' as 'the' and when as 'that'? Two sentences in this skill and have made the wrong call each time. What is the reason for 'den' here if it isn't emphatic?
den + noun = that
den + adjective + noun = the/that
den + adjective treated as noun = the(/that)
"That" was missing as an alternative here because it's a less likely interpretation of the sentence, but it's technically correct.
The use of [den/det] + [definite noun modified by an adjective] is referred to as "double determination" or "double definiteness", and is just how the language works.
Linn has already given the grammatical sense for it and I am not able to expand in that side of things but a tip I use is: Imagine a scene to create context and translate what would seem natural to the scene.
You are correct in saying den is used to emphasise the dirty snow, but unless there are multiple patches of dirty snow with something significant between them, just using "the" is enough to emphasise dirty, as opposed to clean, snow.
It seems more natural though that the speaker is concerned about the child being in any dirt than "that specific patch of dirt." If they were concerned about a specific patch kf dirt and needed to use that, they would probably need to define it more, "that really dirty snow." or "That poisonous dirty snow."
Kinda rambled a bit but I hope this breakdow of the example helps pass on the tip.