After purchasing my home, can I increase my mortgage or take out another loan?

Not without permission from the Post Sales Help to Buy Agent. Further advances must be approved by the Post Sales Help to Buy Agent.
Advances to be used for repaying the equity loans in part (staircasing) or full will usually be welcomed and approved. Advances for other purposes will be considered by the Post Sales Help to Buy Agent on a case by case basis (see question below regarding extending or altering the property).
You may be able to transfer your mortgage to another lender taking part in the scheme following scheme following prior permission from the Post Sales Help to Buy Agent. However, you must ensure your new lender is informed that your home is a Help to Buy property with a second charge entitling the Homes and Communities Agency to a share of the future sale proceeds. You should note that not all lenders will accept a remortgage where there is already an equity loan in place.
The Post Sales Help to Buy Agent may decline permission for further advances or transfer to another lender if after assessment they consider you may be putting yourself in an unsustainable financial position.