How to win with a “weak” Knight?

Ideally, you should follow the chain. By utilizing the already captured territory of the Barbarian (in the area with the upgrade that boosts either attack or defense – the Medusa hiring hall), you can track his movements. Then, transfer the army to the main hero, and use other elements of the chain as well. And somehow, you need to organize an expedition from the starting castle to the Necromancer (you'll need a lot of gold for this). There, you can try a trick: almost kill the Barbarian, but buy him off with one dragon, then finish him off with the remaining army. Then, buy the hero and the dragon in the city near the isthmus to the Necromancer, and use the dragon to defeat him.
I'll say this right away: it won't be that easy with tournament rules, because if you buy him off, you can't use that hero on the same day.
You can also play it differently, going in the other direction. Buy the remaining swordsmen and pikemen, upgrade them for free outside, take three archers and swordsmen/pikemen, and attack Dracogard. However, I'm not sure if you'll have enough troops to take it. You could take more than three archers... But even buying just one red dragon in Dracogard should be enough to defeat the entire Necromancer's army (by the third week).