First let me establish my credentials - I work in a Jobcentre.
A condition of claiming Income Support as a lone parent is that you attend and take part in this interview( which is to give you information on how better off you would be in work, what with tax credits,help with childcare costs etc)
If you don 't attend, you may lose benefit.
Having said that, four weeks is a long time to be without any money. I would advise you to contact the Jobcentre to get them to try and get the interview sorted now, it's unreasonable to wait any longer.
Once you've attended your interview, your claim will be processed, and you could request an urgent benefit payment on the day of the interview.