Learn Polish: Ledwo odkryli tę jaskinię, latarki zaczęły migotać. - They had barely discovered this cave when the flashlights started to flicker.

In Polish, the structure ledwo + [past action], [next action] doesn't require a direct equivalent of "when" or "then". The rapid sequence of events is clear just by putting the two clauses side by side. While you could technically add a or jak to start the second clause, leaving them out is very natural and creates a snappy, dramatic feel.

Why is the first verb odkryli perfective, but the end of the sentence uses the imperfective migotać?

The conjunction ledwo focuses on an action that was just completed, so it almost always takes a perfective verb like odkryli (they discovered). In the second clause, zaczęły (they started) is also perfective because the start of the action is a singular, completed point in time. However, it must be followed by the imperfective infinitive migotać because the flickering itself is a continuous process that began right at that moment.

I sometimes hear native speakers use tą instead of tę in phrases like this. Are they both correct?

Strictly speaking, tę is the correct accusative form of the feminine pronoun ta (this), making tę jaskinię grammatically perfect as the direct object. The word tą is the instrumental form. However, in casual spoken Polish, native speakers very often substitute tą for tę in the accusative case. Using tę shows you know the standard rules, but you should expect to hear tą jaskinię in everyday conversation.