I can't help you with specific names of clerical garments or their construction, but I can certainly try to help you out with some period depictions, at least, in the hopes of someone more versed in this specific topic will find this and be able to help you more. Effigies and Brasses is a database for, well... funerary effigies and brasses from the Medieval period: you can filter by the modern nation where the monument is located, set parameters for years, civilian/clergy/martial, and male/female. Here are Clerical monuments from England, 1300-1400, but using the search feature you can change your parameters. The specific positions/titles are sometimes but generally not noted, and many are drawings and rubbings from 19th or 20th century books, but searching for the individuals can at times yield further information. Word of warning that clerical garb of the time did, at times, incorporate swastikas, and several images on that page shows garments in which they are embroidered on collars, so if that is something you would rather not see I would be happy to pull out all of the images that don't include such imagery and link them individually.

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A few period illustrations of clergy: note that the figures depicted below are either Biblical figures or high-ranking members of the clergy, and I couldn't tell you how specifically the garb for a priest would look, but I hope it gives you a general idea.

• In the Apocalypse in Latin from the mid-late 13th century, this folio shows a young man being presented to John: note that this depicts an event a millennium before the manuscript was written, but the depiction seems consistent with what's depicted elsewhere in Europe at the time.
• In this image from the early-mid 14th century Luttrell Psalter, the assassination of Archbishop of Canterbury Thomas Becket (ca. 1170) is depicted, depicted in a lower level of dress than might be worn during services or other events.
• This image from the early 14th-century Peterborough Psalter gives another depiction, though I'm not certain of the context.
• One more: this one from the 1430s Lives of Saints Edmund and Fremund depicts 5 knights in pentinence before the shrine at the Abbey of Bury St. Edmunds after stealing horses from the monastery