I think the existing answer is a touch unclear; every Statutory Instrument can be said to "amend or repeal legislation, with varying degrees of parliamentary scrutiny" because secondary Legislation is legislation, so I wanted to go into a bit more detail about exactly what makes something a Henry VIII power as opposed to some form of secondary legislation that is not such a power.
What is meant by Henry VIII powers?
Henry VIII powers are powers granted by an Act of Parliament to government ministers, specifically allowing them to alter primary legislation. This distinction is important; in effect, primary legislation is law passed as an Act of Parliament. Secondary or subordinate legislation is law passed through the use of other powers, either granted to government ministers by Parliament, or the Queen's prerogative powers.
Most legislative powers granted to government ministers specifically limit them to the use of Statutory Instruments (Orders in Council, Regulations, Rules and so on). For example, the government created The Official Controls Regulations 2019, which is a Statutory Instrument, and a piece of secondary legislation, through the use of powers granted to them in the European Communities Act 1972, Section 2(2):
(2) Subject to Schedule 2 to this Act, at any time after its passing Her Majesty may by Order in Council, and any designated Minister or department may by order, rules, regulations or scheme, make provision—
(a) for the purpose of implementing any EU obligation of the United Kingdom, or enabling any such obligation to be implemented, or of enabling any rights enjoyed or to be enjoyed by the United Kingdom under or by virtue of the Treaties to be exercised; or
(b) for the purpose of dealing with matters arising out of or related to any such obligation or rights or the coming into force, or the operation from time to time, of subsection (1) above;
and in the exercise of any statutory power or duty, including any power to give directions or to legislate by means of orders, rules, regulations or other subordinate instrument, the person entrusted with the power or duty may have regard to the [F4objects of the EU] and to any such obligation or rights as aforesaid.
Further use of those Powers could allow them to amend those regulations, repeal them altogether or create entirely different ones. What prevents these powers from being Henry VIII powers is that they do not allow the government to amend or create primary legislation. They could not, for example, use those powers to alter the European Communities Act itself or any other Act. In contrast, the Nationality, Immigration and Asylum Act 2002, Section 157 (1) and (2) grants to the government the following powers:
157 Consequential and incidental provision
(1) The Secretary of State may by order make consequential or incidental provision in connection with a provision of this Act.
(2) An order under this section may, in particular—
(a) amend an enactment;
(b) modify the effect of an enactment.
(3) An order under this section must be made by statutory instrument.
(4) An order under this section which amends an enactment shall not be made unless a draft has been laid before and approved by resolution of each House of Parliament.
(5) Any other order under this section shall be subject to annulment pursuant to a resolution of either House of Parliament
The key word there is enactment; this is a power to amend an Act of Parliament. Furthermore, it's not even a power to amend a specific Act of Parliament; the government can use this power to amend any Act of Parliament, including the Act that grants this power itself (in other words, they can theoretically use this power to grant themselves MORE powers). That's a Henry VIII power.
How did they come about?
Funnily enough, they've existed at least since Henry VIII. The Proclamation of the Crown Act 1539 grants proclamations by Henry VIII the force of Acts of Parliament. This association with Henry VIII semi-taints them with an aura of illegitimacy; at least initially they came about because the monarch wanted to grant his policies the force of law without having to deal with the Commons. These days however the usual justification is essentially administrative necessity (I.E. there's too much to bother Parliament with and so the government should just be able to deal with it)
Are they inherently anti-democratic, and if so, is it feasible to abolish them, or do they have a place in the legislative process?
This is somewhat opinion based. As you can see in Nationality, Immigration and Asylum Act there often are controls on the use of the powers; Orders made under the powers granted by this Act must be laid before Parliament, who may then decide to reject them. This is a form of positive oversight; the enabling Act specifies that Parliament must approve the orders made under that power. Other Acts granting Henry VIII powers only mandate negative oversight; I.E. Orders made using the powers must be laid before Parliament, but will come into effect so long as there is no resolution rejecting them (even if there is also no resolution specifically approving them).
There is however no requirement for the enabling Act to include such a provision, so there is at least the possibility of absolutely no scrutiny whatever. I would consider that completely undemocratic; but I couldnt find an instance of such an Act, so I suppose the point is moot. As to whether they have a place, I think the answer is probably yes; there IS an argument that government should sometimes be able to amend or repeal legislation. The scenario that jumps to mind is when implied repeal causes complications of some kind, either because it was somehow incomplete (I.E. the new Act did not wholly conflict with the old) or was unintentional or something similar - in that case I can see the advantage in allowing Government to simply fix the problem, provided there was Parliamentary oversight.
The House of Lords semi-recently (2002) published a report titled HENRY VIII POWERS TO MAKE INCIDENTAL, CONSEQUENTIAL AND SIMILAR PROVISION on the use of these powers, their checks and balances and whether they were appropriate in a number of case studies. It's well worth reading to learn a little more on the subject.