Can a high court enforcement agent get a warrant to enter my house?

Yes

An enforcement agent has authority to 'enter premises' only when he has an enforcement power, such as a writ of control, or a warrant of control. Premises also means 'any place', ^[1] anywhere in England and Wales.^[2]

Bailiffs may try to mislead debtors that 'authority to enter premises' means to commit breaking and entering or to 'get a locksmith'.

Bailiffs also mislead that their enforcement certificate by itself is authority to enter any address in England and Wales without permission.

Bailiffs may only enter peacefully, (using non-violent means) the premises specified on the enforcement power document itself, or anywhere the debtor usually lives or carries on a trade or business.^[3]

If an enforcement agent wants to take control of the debtor's goods at any other premises not specified on the enforcement power, or where the debtor usually lives or carries on a trade, then he must apply for that authority separately.^[4]

If bailiffs take control of goods not on specified premises, a highway in England and Wales or any address the debtor does not live or carry on a trade or business, then the owner of the goods other than the debtor may recover damages for the breach.^[5]

Otherwise, if the debtor owns the goods, they may recover damages my making an application.^[6][7]

^{[1] Paragraph 3 of Schedule 12 to the Tribunals Courts and Enforcement Act 2007
[2] Section 142(2) of the Tribunals, Courts and Enforcement Act 2007
[3] Paragraph 14(6) of Schedule 12 to the Tribunals Courts and Enforcement Act 2007
[4] Paragraph 15 of Schedule 12 to the Tribunals Courts and Enforcement Act 2007
[5] Section 3 of the Torts (Interference with Goods Act 1977
[6] Paragraph 66 of Schedule 12 to the Tribunals Courts and Enforcement Act 2007
[7] Civil Procedure Rule 84.13}