Hey, while passing by I admit I only looked at the test plan and tried to get this moving by executing it.
Thanks for hinting at these further things to check ...

I still had the environment around

root@Jdnsmasq:~# apt-cache policy dnsmasq
dnsmasq:
  Installed: 2.86-1.1ubuntu0.4
  Candidate: 2.86-1.1ubuntu0.4

That resolved well, asking the configured dns (8.8.8.8 in my case) and returning a proper answer.

root@Jdnsmasq:~# dig +short A www.thekelleys.org.uk @127.0.0.1
thekelleys.org.uk.
85.119.82.65
root@Jdnsmasq:~# dig +short A www.thekelleys.org.uk @127.0.0.1
thekelleys.org.uk.
85.119.82.65
root@Jdnsmasq:~# dig +short A www.thekelleys.org.uk @127.0.0.1
thekelleys.org.uk.
85.119.82.65

Since the original issue was about repeating queries (in other context and situation) I ran it a few times.
The log (we still have verbose logging enabled from the first test) shows the forward resolving just as expected:
Jan 05 07:32:56 Jdnsmasq dnsmasq[255]: query[A] www.thekelleys.org.uk from 127.0.0.1
Jan 05 07:32:56 Jdnsmasq dnsmasq[255]: forwarded www.thekelleys.org.uk to 8.8.8.8
Jan 05 07:32:56 Jdnsmasq dnsmasq[255]: reply www.thekelleys.org.uk is <CNAME>
Jan 05 07:32:56 Jdnsmasq dnsmasq[255]: reply thekelleys.org.uk is 85.119.82.65


---


Now dnsmasqs version of a static entry

root@Jdnsmasq:~# echo "address=/domain/1.2.3.4" >> /etc/dnsmasq.conf
root@Jdnsmasq:~# systemctl restart dnsmasq
root@Jdnsmasq:~# dig +short A domain
1.2.3.4


---


Since I had that running over night I also see in the verbose logs all kind of expected background action and all that worked as well.
Like:
Jan 05 07:30:43 Jdnsmasq dnsmasq[255]: cached api.snapcraft.io is 185.125.188.54
Jan 05 07:30:43 Jdnsmasq dnsmasq[255]: cached api.snapcraft.io is 185.125.188.59
Jan 05 07:30:43 Jdnsmasq dnsmasq[255]: cached api.snapcraft.io is 185.125.188.58
Jan 05 07:30:43 Jdnsmasq dnsmasq[255]: cached api.snapcraft.io is 185.125.188.55
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: query[AAAA] canonical-bos01.cdn.snapcraftcontent.com from 127.0.0.1
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: forwarded canonical-bos01.cdn.snapcraftcontent.com to 8.8.8.8
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: query[A] canonical-bos01.cdn.snapcraftcontent.com from 127.0.0.1
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: forwarded canonical-bos01.cdn.snapcraftcontent.com to 8.8.8.8
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: reply canonical-bos01.cdn.snapcraftcontent.com is NODATA-IPv6
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: reply canonical-bos01.cdn.snapcraftcontent.com is 91.189.91.43
Jan 05 07:30:44 Jdnsmasq dnsmasq[255]: reply canonical-bos01.cdn.snapcraftcontent.com is 91.189.91.42
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: query[AAAA] api.snapcraft.io from 127.0.0.1
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: cached api.snapcraft.io is NODATA-IPv6
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: query[A] api.snapcraft.io from 127.0.0.1
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: forwarded api.snapcraft.io to 8.8.8.8
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: reply api.snapcraft.io is 185.125.188.58
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: reply api.snapcraft.io is 185.125.188.55
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: reply api.snapcraft.io is 185.125.188.54
Jan 05 07:30:51 Jdnsmasq dnsmasq[255]: reply api.snapcraft.io is 185.125.188.59
...
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 185.125.190.39
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 91.189.91.81
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 91.189.91.83
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 2620:2d:4000:1::16
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 2620:2d:4002:1::103
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 2620:2d:4002:1::102
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 2620:2d:4000:1::19
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply archive.ubuntu.com is 2620:2d:4002:1::101
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 185.125.190.36
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 91.189.91.81
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 91.189.91.83
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 91.189.91.82
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 185.125.190.39
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 2620:2d:4000:1::16
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 2620:2d:4002:1::101
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 2620:2d:4002:1::103
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 2620:2d:4000:1::19
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply security.ubuntu.com is 2620:2d:4002:1::102
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[SRV] _https._tcp.motd.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded _https._tcp.motd.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply _https._tcp.motd.ubuntu.com is NXDOMAIN
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[A] motd.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded motd.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[AAAA] motd.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded motd.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 54.171.230.55
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 34.254.182.186
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 54.217.10.153
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 34.243.160.129
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 54.247.62.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 2a05:d018:91c:3200:5e0d:21a9:26ca:90b5
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 2a05:d018:91c:3200:c8f:1a06:a2dd:450f
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 2a05:d018:91c:3200:2846:99fb:81b6:1e11
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 2a05:d018:91c:3200:d8b6:37bc:63f9:703c
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply motd.ubuntu.com is 2a05:d018:91c:3200:c887:2f22:290f:a7c
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[SRV] _https._tcp.esm.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded _https._tcp.esm.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply _https._tcp.esm.ubuntu.com is NXDOMAIN
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[A] esm.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded esm.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: query[AAAA] esm.ubuntu.com from 127.0.0.1
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: forwarded esm.ubuntu.com to 8.8.8.8
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 185.125.190.23
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 91.189.91.47
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 91.189.91.46
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 185.125.190.24
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 185.125.190.75
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 2001:67c:1562::21
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 2620:2d:4000:1::30
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 2620:2d:4000:1::2f
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 2001:67c:1562::22
Jan 05 07:38:29 Jdnsmasq dnsmasq[765]: reply esm.ubuntu.com is 2620:2d:4000:1::2e


And overall the systems network functions (currently going through dnsmasq for name resolution) seem happy.