Comments from RHB 

I am approaching this as someone who knows very little about the subject. I will give some high-level comments mainly on structure and content. There are copyediting and prose issues as well, but it is probably worthwhile to tackle the general problems before going into specifics. Before I start, one general point that you will see often in my comments is that there are at times new subjects that are suddenly introduced. Although wikilinks help, a clause or a few words of explanation would smooth the flow of reading for a person not familiar with the subject.

Background to the mission
This section seems a little long relative to the rest of the article. Unless there is a significant expansion planned for the rest of the article, I would recommend reducing this section. There are small details that do not add very much to the Augustine story, e.g., the origin of St Martin’s Church, the fair-haired slaves theory, the pope writing to various kings.
Bede is suddenly introduced without explanation.

Arrival and first efforts
The mission turning back to Rome is not explained. Is there a significance to what happened?
It says “…on the one hand… on the other hand…” but the two items mentioned do not seem to be in opposition as expected when this phrasing is used.
Augustine established first at Canterbury. Since he was the originator, it might be good to give a reason why he chose the location.
The Archbishop of Arles is suddenly introduced. Who is he? Was he based in Arles (in modern-day France) or in Kent? The existence of an Archbishop implies a significant Christian population existing on British soil. This then brings more questions to mind.
It says "That Christmas…". But what year is that?
The king became a saint after his death, but the sentence may be in the wrong place. Did the king die shortly after Augustine's arrival?
In the same paragraph, some information of the existence of Christians before Augustine's arrival is given. This info should go before the discussion of Augustine's arrival.
Mellitus and Justus are suddenly introduced without explanation.
The sentence "A theory by the historian S. Brechter…" is either a run-on sentence or is incomplete.

Additional work
After the first sentence describing the two episcopal sees, there is a sudden jump to after 1066 and a discussion of a cathedral. How are they linked? Also what cathedral is it? The description of the archeology seems out of place in the "Additional work" section. Is it meant to show that Augustine managed to build new churches?
If Aethelbert was only the King of Kent, how was he able to summon bishops outside his realm?
There is a sudden introduction of the tonsure.

Further success
Concerning "temples and usages" is that "temples and their use"? The "former… latter" construct does not work very well here.
Concerning the quote "whoever wishes…", who said that?

I would recommend reading the article out loud. I think you will find some sentence structures that seem a bit complex or difficult to understand. Following the corrections, a good copyedit/prose check is needed. I give as an example, the sentence "As to why Pope Gregory…" could be made better as in "Nothing is mentioned in the sources on why Pope Gregory chose a monk to head the mission".

I hope this has been helpful.