To the best of my belief, you can make a necessary journey in an unnecessary vehicle! I did that on occasions all the way through the first lockdown with my Dino, as there was no saving to be made by SORNing a free to tax car.
Unfortunately, we are all running the gauntlet with respect to uninsured drivers on our roads. If insurance companies are looking to reasons for journeys to retrospectively remove cover from drivers on top of the usual reasons of "ne'er do wells" saving high premiums, then we will be further relying on our own comprehensive cover.
From a recent post on the Westfield Sports Car Club forum, 29 reasons you're allowed to leave home in lockdown (England).
1 If it is ‘reasonably necessary’ to “buy goods or obtain services from” businesses that remain open - either for yourself, household members, or a “vulnerable person” or their household member.
2 To withdraw or deposit money in a bank or building society or similar business.
3 To “take exercise outside”, either alone, with members of your household or support/childcare bubble, or with one person from another household. Under this exemption, there must just be two of you in total - though kids under 5 don’t count towards the limit.
4 To attend a place of worship (though they’re only staying open for private prayer).
5 To attend a Remembrance Sunday event.
6 To visit estate or letting agents or show homes; view properties to buy or rent; prepare a property to move, buy, rent or sell; or move house.
7 To visit someone in your support bubble or childcare bubble. A childcare bubble is when two households join together for informal childcare for a child under 13.
8 To pick up takeaway food or drink.
9 To visit a waste disposal or recycling centre.
10 To attend work, education or training, or provide voluntary or charitable series, if it’s not reasonable to do so from home.
11 To provide care or assistance to a vulnerable person.
12 To provide emergency assistance to any person.
13 To fulfil a legal obligation, including attending court or satisfying bail conditions.
14 To access social services, DWP services, victim of crime services, and asylum and immigration services and interviews.
15 If you’re an elite athlete, to continue training or competition.
16 To seek medical assistance, including to donate blood, attend medical trials or take Covid tests or a vaccine.
17 To avoid injury or illness or escape risk of harm.
18 To be with a mother giving birth, at her request.
19 To visit a household, close family member or friend who is being treated in hospital, a hospice or care home (local rules on visits permitting).
20 To attend a support group, including for domestic abuse victims, addicts and LGBT people, or provide or receive respite care.
21 To visit a household, close family member or friend you reasonably believe is dying.
22 To attend a funeral or wake, or burial ground or garden of remembrance, though there are limits on the numbers who can attend.
23 To attend a marriage or civil partnership, but again there are strict limits on numbers.
24 To allow children with separated parents to move between those two parents’ homes.
25 There are also certain exemptions for children in care and those preparing to be adopted.
26 To visit a vet.
27 To walk or otherwise exercise your pet.
28 To return home if you were on holiday before the lockdown came into force.
29 To visit a close family or friend in prison.
Ultimately, if you are still concerned, a call / e-mail to your broker / insurer and recording the response ought to be definitive.