Varies with location. CYC's business model is that they don't own any chargepoints - they just manage them on behalf of the site owners, mostly local councils etc. who installed them with OLEV grants. The site owners set the price per charge, with a wide variety of tariffs, from "free" to hourly rates or per kWh. For sites that were "free", CYC now add a £1 service charge.
The annual fee is purely for the card - what you then pay for use is exactly the same whether you have the card or just use the mobile app.
(just found out the explanation for my failed charge - despite having "auto_renew=on" in the account settings, they failed to collect the renewal fee last year and the card has expired!).
Polar is different, despite the common ownership - their card has a monthly fee, but gives you access to lower tariffs on most of their chargepoints compared to using their mobile app. Polar card gives access to CYC points (but not vice-versa), though the CYC tariff still applies. CYC also run ChargeplaceScotland on behalf of the Scottish Government - CYC card (but not Polar card) also gives access to those points.
Among other operators:
ChargepointGenie/Geniepoint - They will sell you a card for a one-off fee, but also have a mechanism to let you register any other RFID card that you already have - their model is not raising revenue from the card. Their alternative is a mobile-friendly website rather than an app per se. They used to have a myriad of tariffs like CYC, but have gradually herded their site owners onto two sets of tariffs - London/non-London.
PodPoint - again variable tariffs set by the site, but quite often free. No cards (apart from a few legacy locations that haven't yet been upgraded to their new hardware), but choice of app/website and a special scheme whereby it starts charging immediately you plug in, with 15 minutes grace period to sort out authentication (allowing you to eg. move somewhere else for better mobile reception/less rain, or in emergency allowing a small charge if the authentication fails to work).
Franklin Energy/EV Driver - use backend from Fortum (Finland) with roaming agreements to each other and the other Fortum partners in Europe. Again, card available for one-off fee, or use website.
Shell - Marketing directs you towards an app which has a very poor reputation, but they also accept TheNewMotion cards. TNM is a Netherlands-based operator, though they do have a couple of locations in the UK, and they have lots of roaming agreements across Europe. I believe card is free (must get around to applying for one).