But Sefton MBC says their 'very positive initiative' should be promoted to other local authorities as good practice.
Mr McAvoy said: 'I am appalled at this scheme. If it is government funded I question with whom it has been discussed and what view the Electoral Commission takes of it.
'Whilst participation in elections is to be encouraged, this should not be done through bribes to children.'
A Sefton MBC spokesman said: 'The competition is a borough-wide campaign to raise awareness and participation in elections and is about providing a valuable experience to children in explaining democracy and promoting active citizenship.
'The NUT did not contact Sefton before issuing the press release. It is regrettable the NUT chose not to discuss any concerns it may have had before going to the media.
'With there being just three prizes on offer for children for the whole of the borough, Sefton refutes NUT claims that this constitutes a 'bribe' or 'prizes for votes'.
'This initiative is part of a wider citizenship programme the council is developing to engage with young people and reinvigorate interest in democracy. Children are voters of the future and with declining turnout, we felt we had to stimulate their interest.
'This is a very positive initiative and we've had an excellent response. We will be evaluating the effectiveness of the competition and will be encouraging the Electoral Commission to promote it as good practice'.