I`d have thought that all CD players will play CD-Rs. Even my 8 year old Sony 797 plays CD-Rs. Its CD-RWs that most can`t play. Richers do a Philips that will play CD-RWs for £80, but I wouldn`t expect much in the way of sound quality at that price.
Under £200 I`d look at Sony, Denon or Marantz, though its worth looking out end-of-line, ex-dem and seconhand (sorry, previously cherished) stuff. I picked up a Marantx CD63KI Signature for £200 a couple of years ago when it was still £500, simply by buying secondhand from a local Hi-Fi dealer.