What do we call the little cape that just covers the shoulder?

Since 1912 at least, they have been called 'capelets' according to Merriam-Webster online.

capelet : a small cape usually covering the shoulders

From the 14th century the word tippet seems to have been used. Merriam-Webster defines Tippet

2 : a shoulder cape of fur or cloth often with hanging ends

While I Marc Carlson's Glossary of some medieval clothing terms gives a lengthier description. (bolding mine)

Tippet

[Origin uncertain; some suggest identity with OE. tæpped, tæppet, *teped (pl. tæppedu, tepedu) carpet, hanging, etc. = OHG. teppid, -ith, -it, tepid, -it: both ad. L. tapete (-a, -um) a carpet, tapestry hanging, bed-cover, table-cover. But there are great difficulties both of phonology and of sense. Others suggest a derivative of tip] 1. a. A long narrow slip of cloth or hanging part of dress, formerly worn, either attached to and forming part of the hood, head-dress, or sleeve, or loose, as a scarf or the like. Obs. exc. Hist.b. A garment, usually of fur or wool, covering the shoulders, or the neck and shoulders; a cape or short cloak, often with hanging ends. Now worn chiefly by women and girls, or by men as a part of certain official costumes. In many early quots. (omitted here), senses a and b are not distinguishable.

I have seen it called a Shoulder Cape, shown below on a nurse in the Australian Army Nursing Service working dress.

Other times, the cape may fit atop or into a heavy Western or Australian oilcloth coat for the cold and rain, called a duster.

Since I believe this moment in the movie represents quasi-religious garb, I believe pellegrina is appropriate (although that is closely tied with the Catholic religion).

For the secular version, capelet or tippet is appropriate.

A short cape is often called a cowl. Modern "cowls" are mostly overgrown scarves, but if you look up "medieval cowl" you will see shoulder-length cloaks, usually with hoods. Not all of them open at the front, as in the picture, but some do. A poncho is similar, but I believe a poncho is always a single piece in the front, rather than tied. A shawl is also similar, but I would argue a shawl is usually the product of needlework (rather than sewn fabric pieces). A capelet could probably encompass any of these.