Peter I'll reply here so maybe others can join the discussion.
Ukrainian and Polish are often very similar.
Here are some of the names I know, some I found and some I just guess (?)
Synyak - Syniak or Siniak (siny - colour grey-blue)
Makovytsia - Makowica (mak - poppy)
Dil - Dział (divide)
Velykyi Dil - Wielki Dział (great divide)
Buzhora - Bużora
Bystra - Bystra (fast-flowing water?)
Velykyi Sholes - Wielki Sałasz/Wielki Szałas? (what's szőlősi in Hungarian? maybe something like a wooden primitive hut, especially for shepherds or sheep? - Polish "szałas" or in highlanders' dialects "sałasz")
Vyshkovo - Wyszkowo
Polonyna Rivna - Połonina Równa (flat mountain)
Ootra Hora - I think there's a mistake here, should be Ostra Hora - Ostra Góra in Polish (sharp mountain/peak)
Lyutans'ka Holytsia - Lutańska Golica? (golica - probably from 'goły' - naked, not covered with vegetation)
Polonyna Borzhava - Połonina Borżawa
Kuk - ??? (means 'hip' (part of the body) in southern Slavic languages, also often used in mountain names, e.g. Bobotov Kuk in Durmitor in Montenegro)
Bozheva - Bożewa? (something like 'God's place'?)
Polonyna Krasna - Połonina Krasna (Red Mountain?)
Syhlans'kyi - Sichlański? (swampy place? - I found that 'sichła' means swamp in some dialects, it's found in some other local names, e.g. village Murzasichle in the Tatras)
Menchul - Menczuł/Menczył/Munczeł
Klymova - Klimowa
Svydivets' - Świdowiec
Blyznytsia - Bliźnica
Chornohora - Czarnohora (black mountain)
Hoverla - Howerla
Brebeneckul - Brebenieskuł
Pip Ivan Chornihors'kyi - Pop Iwan (Czarnohorski) (Orthodox priest Ivan)
Petros - I think it should be spelt Pietros (Polish name is the same, perhaps a variation of your name - Peter
)
Hutyn Tomnatyk - Gutin Tomnatyk/Gutin Tomnatek
Rebra - Rebra
Turkul - Turkuł
Maramoroshs'kyi masyv - Góry Marmaroskie
Pip Ivan - Pop Iwan (Marmaroski)