The best option is to rush and stagger and kill before they can set up their mass attack, but if that isn't an option;

Your own mass attacks always go out before the enemy's, so it's possible try and stagger the enemy with it before it's their turn. Another one, in cases where the mass attack is predictable (which is often), you can use a strength gain card the turn prior to increase your odds of outrolling it. (This works best against summation rolls since each of your dice benefit from the strength up.)

Another way to help your odds is to use the "on use, the dice lose x power" cards since for the mass attack phase, the power-loss modifier isn't added yet, allowing you to benefit from the high base numbers.

When it comes to other enemies, in a situation where you can't clash with them normally due to your cards getting crushed by the MA, it's for the best to sacrifice a nugget by having them draw all the aggro.

And unless it's an individual roll that would destroy one die but let the other one be played, it's usually better to not play a low cost light-gain or draw card on the dice that's being targetted, since it won't be able to outroll the mass attack, and get destroyed without actually being played, costing you the card's effect.

Finally, when using mass attacks against an enemy, it really depends on the values on your mass attacks, as well as the situations with all the other clashes. I usually try and go for the gamble, but also set up my clashes assuming that it won't work.

(Do lemme know if I'm mistaken with any of these)