Can New Member States Be Admitted to the EU with an Ongoing Territorial Conflict?
I think yes if by "Ongoing Territorial Conflict" you mean "territorial disputes" (as you seem to) rather than open warfare. Spain was admitted (in 1985) to the EU while having a territorial dispute with the UK over Gibraltar (still ongoing). The UK was in the EU then (since 1973).
Greece (EU member since 1981) had & has territorial disputes with Turkey, which is not an EU member, but also started accession talks (now frozen). I'm not entirely sure when those disputes started, but they've been going on at least since the 1970s.
As for your own example, Cyprus, since it is a more recent accession, we have more of the documents online. I'm not entirely sure what is the most relevant bit, but probably from the Cyprus accession protocol:
Article 1
The application of the acquis shall be suspended in those areas of the Republic of Cyprus in which the Government of the Republic of Cyprus does not exercise effective control.
The Council, acting unanimously on the basis of a proposal from the Commission, shall decide on the withdrawal of the suspension referred to in paragraph 1.
[...]
Article 4
In the event of a settlement, the Council, acting unanimously on the basis of a proposal from the Commission, shall decide on the adaptations to the terms concerning the accession of Cyprus to the European Union with regard to the Turkish Cypriot Community.
There was actually a pretty formal settlement attempt as the Annan Plan. This actually failed with the Greek Cypriot side rejecting the proposal in a referendum in 2004, but nonetheless, EU accession was not affected, completing in the same year. And FWTW, much more recently Erdogan too has been rejecting other similar attempts at federalization of Cyprus, asking for a straight two-states solution but "adding that the Turkish north is the “apple of Turkey’s eye.”" That (July 2024) piece also mentions that
Multiple attempts to find a compromise settlement over the years have failed, the last one in 2017 in Crans-Montana, Switzerland; and formal talks have not resumed since then.
As you seem to intuit by referring to Cyprus foremost, their situation of not having full territorial control is indeed perhaps the most relevant to e.g. Moldova or Georgia, and perhaps even Ukraine.
The only thing that I think hasn't been done in terms of precedent is admitting into the EU a country still fighting an active war, rather than this kind of "frozen conflict".