sudo apt-get install python-poppler-qt4
Reading package lists... Done
Building dependency tree
Reading state information... Done
Starting pkgProblemResolver with broken count: 1
Starting 2 pkgProblemResolver with broken count: 1
Investigating (0) python-poppler-qt4 [ amd64 ] < none -> 0.24.0-1build1 > ( universe/python )
Broken python-poppler-qt4:amd64 Depends on sip-api-11.2 [ amd64 ] < none -> > ( none )
Considering python-sip:amd64 4 as a solution to python-poppler-qt4:amd64 9999
Considering python-sip:amd64 4 as a solution to python-poppler-qt4:amd64 9999
Done
Some packages could not be installed. This may mean that you have
requested an impossible situation or if you are using the unstable
distribution that some required packages have not yet been created
or been moved out of Incoming.
The following information may help to resolve the situation:

The following packages have unmet dependencies:
python-poppler-qt4 : Depends: sip-api-11.2
E: Unable to correct problems, you have held broken packages.

Using xenial 16.04, KDE neon LTS User Edition 5.12.

On the one hand you could try chasing down the dependencies to see about getting python-qt4 and needed dependencies installed (have you tried installing/investigating sip-api to see why that's failing?). On the other hand, I wonder why you couldn't use the qt5 version? (granted it would take some time to convert things from qt4 to qt5)

I am actually trying to install Frescobaldi (http://frescobaldi.org/) and it requires python3-poppler-qt4. But python3-poppler-qt4 is failing on sip-py3api-11.2 which appears to be a package which is no longer available. From what I can tell, python3-sip has replaced sip-py3api-11.2. But python3-poppler-qt4 does not appear to recognize python3-sip.