Can I say "Please give your children big cheers at the playing field? Or should I say "Please give your children a big cheer?" Do these mean different things?
"A big cheer" is correct in British English, meaning one applause (of course there is usually not only one clapping of hands in an audience - except at a concert, when somebody thinks the piece is over, when there is only a pause - , "applause" being a singular for a general approbation), or one (general) shout of encouragement.
There are a number of expressions in use: 'Please give a big cheer for ...';'cheer loudly for...'; 'let your cheers ring out for...'. The other one you mention 'Please give big cheers for...' is not normally used, though it is perfectly correct grammatically, and everyone would know what you meant. These things simply boil down to a question of usage. Some things are used others are not, and the only way to find out is through long experience. I know people who have lived in Britain for decades but still say things in a way which everyone understands but which reveals that English is not their first language. That particular expression would fit into that category.