the notes are e 6 string open [but not played, but which can also be fretted to make a nominal g fifth fret, 5 string 3 fret up from partial capo, which is nominal c, 4 th string 2 fret up from partial capo which is nominal e,3 string open which is nominal g, 2 nd string 3rd fret which is nominal c, this is a commomn c shape, however as you do not like the open top string which gives us a nominal c9, you can try first string fretting with 5 th fret with pinky finger, this gives you a nominal g note, that is a c major shape, however because of the partial capo, you are in fact playing a d chord.
when i am playing like this i am thinking as if i am in g major c major and d major ,
however i am really in a major, d major, and e major. now i am not a basher or strummer of the guitar but a finger picker, with finger picking it is not necesaary to play all the strings.
so i might pick just the middle four strings 5432 of the nominal c chord[which in fact is tonally d major] or if i want to play 6 string i move my finger from the fifth to the sixth[5 frt], and fret the first string on the fifth fret with my little finger, as i do not play all 6 strings at once this works.
i suppose it depends upon your style of playing what i advocate works for finger picking or even for advanced plectrun playing where the plectrum imitates finger style. it may not work if the player wanst to strum or whack all 6 strings together