No searchhelp for a particular techname: I have an InputField defined as follows:

InputField (10,1) "Weighting" (10,20) size=2 name="z_weighting" techname="MPOP-GEWGR"

Why don't the F4 values appear when we press F4 on this input field?

There are many SAP fields that do not have a standard searchhelp defined in the repository (transaction SE11). The F4 search help in such cases is directly implemented in the SAP application; in most cases the application calls an ABAP function module.

You need to implement  an ABAP exit program for the GUIXT F4 searchhelp:

Step 1:

In order to obtain the name of the function module that implements the searchhelp, use Tips&Tricks Nr. 18. The called function is shown in the ABAP "call stack" when the searchhelp is opened up.

In your particular case you will find that this is the function "MPOP_GEWGR_HELP".

Step 2:

Implement the exit as described in the "InputField" keyword documentation.
See also Q&A InputAssistant Nr. 317 for an example.