for example, a pastoral mission

Diocesan
which means "of or concerning a diocese"
I would associate pastoral with "of or relating to a pastor", "relating to a teacher's responsibility for the general well-being of students" or "relating to or associated with shepherds or flocks or herds"

Sorry, it's definitely not diocesan (Diözesan-), that's a completely different word.

'Pastoral' is in the ballpark, but it would more often be understood by the general public as referring only to the work of the ordained clergy. Even in the broader sense, pastoral care normally means one-to-one contact with individuals, such as visitation (of the elderly or hospital patients) and counseling (therapy, advice, psychological help).

'Charitable' is rather old-fashioned in tone but still often used. A clinic, school, orphanage, etc. that provides free services and is supported by donations is a charitable institution.

'Diakonie/Diakonisches Werk' of the church in more general terms is often 'outreach' in the US in a domestic context, that is, in the church's own neighborhood/city/region/nation. Terms such as social service(s), community service(s), social welfare, etc. would also be understood but may be more typical in secular contexts. International outreach could also be called relief work, humanitarian aid, etc.

'Mission(s)' is an older word still occasionally seen in this context; in the US it tends to be used nowadays more in the context of evangelism and missionary work, that is, not only trying to help people but persuading them to convert to Christianity, especially overseas. A 'diakonisches Jahr' could be called a missionary year if it's overseas, a community service year if it's local, or perhaps simply a year spent doing volunteer outreach or social service.

  related discussion:Diakonie
  related discussion:Diakonisches Werk
  related discussion:diakonisch

I also found this thread (=forum001_unsolved_e&file=20050729102010) via Google but the link seems to be broken:
  related discussion:diakonisch


Related words have to do with the office of deacon in a church (Diakonen-/diakonal = diaconal):
  related discussion:Diakonat -- diaconate

Wow, the links all work and they all have names! Cooooool .... (((-:

Sorry, I realized I didn't completely answer the original question. 'Diaconal' refers only to the ecclesiastical office of deacon, and 'diaconic/diaconical' are either non-words or extremely rare, so I would definitely not recommend using any of those.