Hello Petr,
This looks like a "double hop" authentication problem (see https://techcommunity.microsoft.com/t5/ask-the-directory-services-team/understanding-kerberos-double-hop/ba-p/395463).
Impersonating the client on the ("first") server allows resources local to that server to be accessed, but there are probably no credentials available to authenticate with a second server. The link above contains a more detailed description of the problem and a potential solution ("delegation").
Gary