A referral from a weak connection can still help route or flag an application, but its credibility is naturally lower than a referral from someone who genuinely knows your work.

Do not exaggerate the relationship. If the person is comfortable referring after the conversation, let them decide what they can honestly say.

The referral should support a strong application, not compensate for one that is not ready. If useful, this guide on consulting referral guide gives more detail.

weak referral still better than random cold app imo, assuming they actually offered.

I wouldn't wait forever for a 'perfect' referrer if app is ready.