Hi Jill If you are supplementing, you are adding to something - in this case, it is probably a hotel bill. The sentence is saying that the restaurant is able to serve breakfast and dinner, but you will have to pay extra for them because the price you have been quoted for the room / hotel / holiday does not include breakfast or dinner. You pay a certain amount for your hotel room, then each additional optional service that you pay for is called a 'supplement'. The term is frequently used in holiday brochures.

I think Neal's right. It must be a british expression I don't recognize. The usual phrase in US english would be 'additional charge' or sometimes just 'extra'. 'supplement' seems wrong because it indicates something extra you are getting, not being charged. In other words, 'supplement' means positive, 'charge' means negative. I can understand a hotel (maybe using british english?) calling extra add-on goodies 'supplements'. But if they say 'at a supplement' that seems like inappropriate meaning. Not quite wrong, but not quite right. I think you can understand the meaning, but I wouldn't suggest you speak so unclearly yourself. One thing in american culture for sure is that it's considered weak or annoying (or old-fashioned) to hide behind vague wording. If you are going to charge the customer extra, just say so. Don't hide behind unclear words.

supplement here would mean 'supplementary charge' which means additional charge this is in contrast to complimentary as in 'complimentary breakfast' which means the meal was already included in the room charge