Now it works!
I installed linux again, started from begining.
Hiawatha was there, operational, reading /etc/hiawatha/hiawatha.conf.
Then I installed PHP from menu / setup / puppy package manager (it also needs libmcrypt, net-snmp, t1lib packages).
Into hiawatha.conf I only added
CGIhandler = /usr/bin/php-cgi:php
StartFile = index.php
and Interface = ip_address under Binding {}
Simple as it should be.
Problem is in LHMP, do not install it, if someone also complains, allways first ask if LHMP is installed!!!
Hiawatha in that package is reading from /etc, not /etc/hiawatha, so you should copy hiawatha.conf in /etc but who knows what else is corrupted. PHP files are nowhere to be found, not in /usr/bin/php nor anywhere on disk drive. When I install PHP only from link under this text, I can see that it is installed in puppy package manager, but there are again no PHP files on disk.
problematic LHMP is LHMP-8.0-5.3.4-5.3.10
problematic PHP only:
http://www.murga-linux.com/puppy/viewtopic.php?t=95479&sid=0880a4da6a56ab3c809d6d466171167eSo to conclude, best option is to use allready installed Hiawatha, and to add PHP as a puppy package manager packet. Then add those 2-3 lines of code to /etc/hiawatha/hiawatha.conf