Several different things to consider here:
If you have children from a previous relationship, then you need to give them bullet-proof security. Which means not having anyone else financially involved in their home. How would you feel if you split up after giving DP a share in the home, couldn't afford to buy him out, and had to move you and the children out to a cheaper place?
Being on the deeds (ownership) is not the same as being on the mortgage, although the mortgage company will have a say in who is on the deeds.
e.g. they may ask you to pay an insurance fee for their benefit.
If he is not on the deeds or mortgage, it is perfectly fair that he does not pay for the mortgage, or any renovations or extensions or big bills like boiler repair. Those should be entirely paid by you.
He should be paying a fair share of household bills only, not mortgage.
However, as he is essentially living rent-free, you and he might agree that he pays you a small amount of rent.
There are good legal reasons for calling his payments 'rent' and not 'share of mortgage' - if he can prove he has been paying your mortgage, it can get very messy if you break up.
Or, you might decide it is fairer for him to not pay rent to you, but instead put that money away each month in long-term savings or investments, for his own benefit, as an alternative to building equity in the house. He could even get himself a buy-to-let.
Eventually he could buy a share of the house from you with a lump sum, e.g. when your children have left home. The split can be specified legally, e.g. 70/30 or whatever, not equal shares.
What is not fair is for him to pay "into the household" without being clear if it is for your mortgage or for rent.