I've checked my main Boer war database and I have little info, definitely nothing picked up from the Advertiser, though its always possible it was missed. The Rhodes family I am familiar with are the ones connected with the York and Lancaster regiment so I've done a bit of digging as I would have begun tracking W M Rhodes down eventually anyway.
There was a Lt W W Rhodes on the ship "India" which sailed 23rd Jul 1903 from London to Australia, contracted to land at Bombay, plus there is a death recorded in the Royal Artillery Officers Deaths (via FMP) for a Lt W McT Rhodes on 28th Jun 1904 - checking around this date in the Rotherham Advertiser and South Yorkshire Times might provide an informative obituary. He died at Allahabad, India.
A birth is registered In Rotherham 1876 Q1 for a WILFRED McTURK RHODES, and Wilfred M appears in the 1881 and 1891 census at "Woodlands" on Doncaster Road, Rotherham. He was the son of James Rhodes, steel manufacturer and wife Helen. I think this man is a likely match especially as there weren't any other likely candidates.
I've started making use of the newspaper collection on FMP to search for the men I am having trouble identifying - the Sheffield Telegraph and Independent are available so they can provide a useful shortcut. Rotherham archives have a collection of Sheffield Telegraphs as well. There is probably quite a lot of information on the Rhodes family in general available - might be worth checking the Rotherham archives catalogue online.