I'm not too familiar with the 1.6 Pinto engine (also called the Kent engine and found in the Cortina). That said, I see nowhere that a push rod would be supported and feel 99% confident it does not have one. The pump drives off an eccentric on the camshaft. That would be the silver metal area you see in the picture.

 If possible turn the engine by hand and watching with a flashlight see if that eccentric is not worn away (it should look like a bump). If possible try and get the raised spot pointing towards the area the pump lever would contact it. Then try installing the fuel pump. You should feel resistance because that 'bump' is what makes the pump lever move. It there is no resistance then the eccentric isn't making contact with the pump. The cause could be the eccentric is worn or the pump is not correct. 
Update:  I magnified the image and it is hard to tell if there is a build up of grime on the eccentric, or there is excessive wear, or there is a designed in channel in the eccentric to stabilize the pump lever.

It could also be that the pump is fine but there is either a clog or a leak somewhere that is causing the pump to draw air. I'm doubting either because you state manually the pump does work when actuated by hand.

  If for some reason the pump can not be made to function there are workarounds. Electric fuel pumps are available all over Ebay for about $15. They will need 12 volts to run but I assume you have a battery/alternator on this engine already. Just make sure you have a switch wired in so you can easily access it and shut the fuel flow off in an emergency.  This is also necessary when you stop the engine or it stalls. The existing mechanical pump opening can just be filled with the old pump, or as is commonly done, a block off plate can be made to cover the opening.

As with all modifications: Do at YOUR Own Risk.