The question as posed needs a little bit of unpacking.
First impression and the likely conclusion based on the following logic is NO, the clerk cannot object to someone applying for a post.
But we have a road to travel before we get to NO.
Can 'someone' object to 'something?'
Yes of course they can - we are all free to form and voice our own opinions within normal qualified entitlements. The key unasked part of the question however is what relevance, importance or credibility do personal opinions have.
The clerk might express an opinion that they don't want a certain individual doing a certain task - for that opinion to have any relevance, importance or credibility it would need to be quantified and compliant with legislative requirements. EG a clerk 'could' say, I don't like that person because...[insert valid reason]. They could [should] not say I don't like that person because they are ...[insert protected category]. That would be unlawful. But it is simply a personal opinion and it should have no sway upon a process.
We've probably all had bosses that we don't like or don't get on with. It doesn't infer any right, entitlement or authority to have the boss removed though - unless there are justifiable reasons. We get on with it.
In this instance, if we assume there is a process of requiring someone to fill a post and then someone applies for that post it is for the selection process to determine who gets the post - NOT the clerk based on a personal opinion.
It seems to be another example of people not really grasping the master / servant relationship of employer / employee.
It is for the council - or a suitably delegated part thereof - to make and implement decisions.
Can the clerk object to someone applying for a position - well, they can express a personal opinion if they like, but unless they are the delegated decision maker it makes absolutely no difference whatsoever - it probably highlights deeper issues if they do though.
What is this obsession with clerks 'raising a grievance?' What would they have to be aggrieved about? They don't like the new boss? Unless there are GROUNDS for a grievance there is no grievance - just a clerk behaving inappropriately and unprofessionally.