The EGR valve and metal pipe connects from the exhaust manifold (header) to somewhere on the intake manifold below the carb.
In an earlier thread you showed a picture of what someone identified as a 'heat shield'.
In this heat shield there is a large hole, through which the EGR metal pipe usually screws into either a threaded hole in the exhaust manifold or into a threaded hole in a plate that is attached to the manifold by 2 bolts. (the previous owner could have blanked off this threaded hole when he removed the EGR)
What I've identified as an EGR valve and pipe in a previous post
I believe should connect to the exhaust manifold via the threaded fixing on the metal pipe end - similar to what is shown in this picture :-
What I've identified as an 'air injection check valve and hose'
should connect to the air injection manifold on the cylinder head - this manifold is a metal pipe with 4 smaller metal pipes that connect to drillings in the cylinder head leading to the exhaust ports. This manifold afaik is situated between the spark plugs and the exhaust cam box i.e. in the space between the 2 cam boxes on the cylinder head. I suspect that the threaded fitting on the check valve should screw into the air injection manifold. If it does fit in this way, will the hose end reach the outlet on the air pump? Although you seem to be missing the 'diverter valve' and the diagram shows the diverter valve fits between the air pump and the check valve i.e. the order is air pump>diverter valve>check valve>injection manifold on cyl. head.
But as
@Slotman has suggested, there were different emissions/anti-pollution equipment used on different years of the 124 Spider so it's important to follow the correct diagram and use the correct pics for your exact model/year of car. Be aware also that the diagrams/schematics that Fiat used in their manuals don't accurately reflect what the individual components look like in real life nor their exact locations relative to other components (e.g. pipes/hoses can be longer or shorter that shown), so these schematics are only a guide or perhaps were used as a teaching aid in their training courses.
I'm repeating the pic of under the hood of a 124 Spider with arrows to try to clarify the above suggestions:-
Blue arrows indicate the EGR valve and metal pipe connected to the exhaust manifold
Green arrow shows where the air injection check valve is attached to the air injection manifold on the cylinder head
Yellow arrow shows a cap/cover on the hot air take-off on the exhaust manifold heat shield - disregard this, it's not stock, presumably someone fitted it when the stock air cleaner unit (which usually has a hot air pick-up connection pipe to the exhaust manifold) was removed to fit a 'sporty' air filter.
I'm finding it difficult to try to advise you when I can't see what you have on your engine e.g. there were at least 4 different styles of exhaust manifold used on the 124 Spider, some had 2 outlet pipes, other only 1, some had the EGR connection on top, others had it semi-concealed under the manifold etc. some cars had catalytic converters, others didn't, (afaik, the cars with catalytic converters all had 2 red warning lights on the centre dashboard panel - 1 light indicated the cat. was overheating, the other possibly indicated the cat. had to be replaced - they had a finite lifespan, or maybe one of these red lights was indicating that the EGR system needed to be serviced, I can't recall for sure).
In order to try to help you any further, could you post pics of the following - exhaust manifold, intake manifold, base of carb and any vacuum connection ports, top of cylinder head showing between the cam boxes, any unconnected pipes/hoses around engine or elsewhere in engine bay, any unconnected units plus any other possible emissions- related parts that came with the car - I'm not seeing e.g. the air diverter valve or some of the EGR pipework.
