To the best of my knowledge, no credit card offers Avios (as Air Miles are now re-branded) with cash withdrawals.
Further (again to the best of my knowledge) you'll always have to pay a fee (and often a higher rate of interest as well) when you withdraw cash.
To make things even worse, cards that offer Avios are often those that charge an annual fee.
For example, the TSB Premier Avios Credit Card costs you £50 per year just to have it in your pocket. Using your example of withdrawing £200 cash at a time, you'd have to pay £6 for each withdrawal (because the charge is 3%, subject to a minimum of £3) and you wouldn't get any Avios anyway. [The only good news is that the same, quite high, rate of interest applies to cash withdrawals as it does to direct purchases; many firms charge significantly higher rates for cash withdrawals].
Quote:
"Eligible spend does not include cash withdrawals, money transfers, cheques, interest, fees or charges, Payment Protection Cover, transactions at casinos, betting outlets or bookmakers or payments to offline or online current, savings, sharetrading, spread betting, gambling or any other similar account."
Source:
http://www.abouttsb.co.uk/credit_cards/landing_pages/premier-avios.html?WT.mc_id=109335730|282481341|0&so=GOV_VAVP_DESK