Sintendo - it's bad that the psych wasn't there or couldn't send someone in his/her place. Was your P's GP present.
There are clearly some concerns about the safety of the baby and that is why they are saying your P should not stay overnight. The child protection plan should be clear, and state exactly what arrangements are to be made to ensure the safety of your baby. A core assessment is a social worker making investigations into your circumstances, your ability to parent your child in a safe way etc. and this assessment will obviously include details of your P (length of your relationship) and your views on the concerns that have been raised by the conslt psych etc etc. A social worker will come to your home to carrying out this assessment. Just be honest and tell them how you feel about your relationship with your P, whether you have any concerns, e.g. have you seen him in a delusional state related to his mental illness.
There is clearly no concern about your ability to parent, it is your relationship with the baby's father that is the concern, raised as you say by the conslt psych whose view will carry a lot of weight. Social workers are not going to take any risks if a person has been diagnosed as suffering from paranoid schizophrenia by a Conslt Psych. His GP should be advised of any diagnoses that are made, so maybe it would be helpful for him to go and see his GP to find out exactly what his diagnosis is.
A core group is when there is a child protection plan in place and the relevant people (including yourself and your P) meet at frequent intervals to check on the suitability of the plan and whether any changes need to be made.
I appreciate you are very vulnerable at this late stage in your pregnancy, but you must be more assertive with the social workers - they do tend to jabber on as though parents understand and forget that what they do every day is completely new for parents. If there is anything you don't understand (and there clearly is) you must clarify what they mean. I know it's difficult when there are up to 10/12 strangers sitting around a table, but you must speak up. If not then, get hold of the social worker after and get her/him to explain in plain english what they are saying.
It sounds to me that they are trusting you to supervise the contact between your baby and your P and not leave the baby alone with him, and not allow overnight stays. The fact that you are intending moving nearer to your partner won't go in your favour, but on the other hand, if the plan is that you are supervising all contact, then where you live shouldn't really be an issue.
I would caution once again to co-operate with the social workers and not to minimise your Ps mental health difficulties and use of heroin (I assume) in the past. Many people with mental health problems self medicate in an attempt to make themselves feel better, but of course that causes 2 problems, mental health problems and addiction to a particular drug(s).
As I'm sure you are aware a new baby can be a very stressful time for parents and you can'tknow yet whether this might add to your Ps mental health problems. This is why the social services will be wanting to assess over time whether the baby is going to be safe if in the future you decide to live with your P. They will be cautious at first, but if things go well, hopefully they (and you) will be satisfied that the baby is not at any risk from your P.
Happy to help further if necessary. Am a retired social worker and manager (25 years for a LA involved in CP, fostering and adoption)
Sorry Arcadia - I wasn't meaning to dismiss legal assistance "out of hand" far from it. I was just being factual. It is as you will know the duty of the social workers to ensure the birthparents are legally represented in any proceedings.