I tutor in Surrey and charge £30 minimum (to children of friends), up to £50. I get fast results and have to turn away a lot of pupils. Obviously, given my line of work, I would say this but: no, don't drop your rates. You are a highly experienced professional. Your teaching skills and your ability to get quick, long lasting results from your pupils are what parents are paying for.
Even if you were working 7 hours a day, 5 days a week for 52 weeks of the year, which as a tutor, of course, you don't, you'd still only be bringing in £45k on £35 ph. That's not a very high salary for a qualified professional with 20 years experience.
And as you must factor in freelance conditions (no sick pay, holiday pay, NI contributions from employer, employer pension scheme, liability cover etc) then of course your rates must reflect this.
Avoid people who want tutoring for rock bottom prices. People who value their children's tutors also ensure the children turn up on time, do their prep, don't miss sessions. Hagglers tend to have very weak attitudes generally to the part they play in supporting their children's tutoring.
We pay a local tutor £35 ph for our son and insist on paying her if we have to miss a session due to social functions. Our choice, not hers, if we're away for a weekend. She's excellent, we respect her. Why would we want a cheaper, less effective tutor or one resentful of being driven down to bargain rates?
Cultivate clients who respect your worth and what you do.