I've added an untagged way way/1429277102 and added it back to each of the relations. This possibly wasn't the most efficient way of doing it, but it did work.
Unfortunately, the iD editor is not good at spotting gaps. You can use overpass to detect problems - see https://community.openstreetmap.org/t/finding-broken-relations-with-overpass/129031 - and you can also use Josm's validator to do the same thing: @SomeoneElse/diary/406398 .

There were 27 relations in total. I added them back piecemeal because other people were fixing other errors in the relations at the same time.
I've not commented on the changeset that introduced the gaps (changeset/171781477 ) because it's their first change and it wouldn't be fair that their introduction to OSM was the mess of relations that is West Dublin.

Hello,
If way/233317913/history has a sidewalk on both sides the normal tag for that would be something like "sidewalk=both". "footway=sidewalk" notmally means "this is actually a footway of some other road".
Would it be OK to change this to "sidewalk=both" (if that would be correct)?
Best Regards,
Andy

Hello,
Along with quite a few other relations, https://osm.mapki.com/history/relation/2267444 seems to have been broken by your edits in the last day or so.
Previously it looked like this:
https://overpass-turbo.eu/s/2beU
and now it looks like this:
https://overpass-turbo.eu/s/2beW
An easy way to check for gaps in relations is, for every relation that you edit, go to that relation's OSM page (you can click through from your changeset). Here that would be relation/2267444#map=15/52.67545/-6.84360 - you can see (until somone comes along and fixes it) an obvious gap there.
Best Regards,
Andy

I think some of the separately mapped sidewalks in here are a bit incomplete - see for example osm.org/directions?engine=fossgis_valhalla_foot&route=53.634039%2C-2.815702%3B53.634007%2C-2.815523#map=19/53.634189/-2.814973 .
If it was me, I'd just use "sidewalk=left" or similar here; if you're going to use separately mapped sidewalks you need to make sure that you join everything up.
Best Regards,
Andy

After editing any paths involved in a hiking route relation (e.g. all the coastal ones!) it makes sense to check http://ra.osmsurround.org/analyzeRelation?relationId=77964&noCache=true&_noCache=on (where 77964 is the relation) so that it matches what is signed on the ground. There are obviously genuine duplications (high and low tide, whether paths across the ranges are open etc.) but there are some very dubious spurs still.

I've fixed a dodgy overlap at the north end of that which clearly didn't exist.
I've assumed that the inland path to relation/77964 as I presume that that is where that is signed and routed now (my GPS traces here are from 8 years ago and may not be accurate any more).

Hello - it looks like this edit introduced a small spur to the Wales Coast Path - I've split way/1390827060 so that hopefully it's contiguous again.

Hello,
It looks like way/1379326419 had been accidentally missed from NCN4 - I've added it back.
Cheers,
Andy

The woodland at way/148883959/history got deleted in this changeset. What were you trying to do?

Any idea what "building=s" is on way/252175919#map=19/53.269303/-3.914435 ?

What sort of building is a "seb"? way/1367284049 .

Hello,
Lots of the houses in this changeset are "building=semidetached_house100752" - I suspect that is a typo?
Regards,
Andy

Some of these building values were odd - I changed them in changeset/165005490 .

Oops - looks like this added an extra bit to NCN425 4247567 here - I've removed it from the relation.

This changeset broke the Centenery WAy at changeset/163340221#map=19/52.502942/-1.472353 . I've fixed it.

Oops - way/1363664264 wasn't joined to the service road here.
I also filled in the gap in the Centenery Way.

Hello,
There seems to be a bit of an odd spur on a few route relations at relation/1832946#map=19/51.626375/-1.013466 . Is that a mistake and do they all go to the south, or is there a bit through the field missing?
Affected relations can be seen at https://map.atownsend.org.uk/maps/map/map.html#19/51.62638/-1.01347 .
Best Regards,
Andy

It looks like one of these changes broke at least one relation - "12458163 | Kilconny ED". I've fixed it in changeset/163005426 - does everything still look OK?

Hello and welcome to OpenStreetMap!
I think that this edit might have introduced a gap in the Saxon_Shore_Way relation 8864. I've filled it in now:
relation/8864#map=18/51.384666/0.523991
It might also be worth checking the other bus route relations - way/1359466727 is in fewer than the way to the south.
Best Regards,
Andy