I've been doing specimen MAT admission test - but I couldn't figure out the answer to the parts III, and IV of question 6 (https://www.maths.ox.ac.uk/system/files/attachments/speca.pdf). Is there some kind of a trick?

As for part III you only need to use the process of elimination. Notice, that there are only three possible situations, since at east one of Alice, Bob, and Charlie has a white hat; namely: one of them has a white hat, or two of them have, or all of them have white hats. Let's consider each case:

Case 1) only one person has a white hat - it's impossible, because then, let's assume without loss of generality that it is Alice who has the white hat,  Alice would see two  black hats, so since there must be at least one white hat she would know, that her hat is white.

So what we can deduce is that at least two of them wear white hats.

Case 2) If Alice and Bob both have whit hats, they will see Charlie in a black hat, and the other one in a white one, so they will have to decide between case 2 and case 1, so they will answer no, as they cannot know what colour their hat is. Charlie on the other hand will see two whit hats, and will have to decide between case 2 and 3, as they all may have white hats, or it's only him having a black one.

Case 3) They all will face the same decison as Charlie in case 2.

So both cases 2 and 3 are possible.

Now for part IV you could again use the process of elimination, or use a trick. 

The trick:  Notice that you have to use your knowledge from part III, notice further, that at most one of them has a black hat and that only Alice said something else than the other two, so they have different hats, as their answers differed. Recall there is at most one black hat, so Alice must have it.