Do associated companies affect my corporation tax rate?

Yes, if you have associated companies, generally companies under common control, the £50,000 and £250,000 profit thresholds for the small profits rate and marginal relief are divided between them. So owning two associated companies each with £40,000 profit could mean you no longer qualify for the full 19% rate on both, because the threshold is effectively halved. The rules for what counts as an associated company can be nuanced, so it's worth getting an accountant to check your group structure and confirm how it affects your tax rate.