I received multiple transactions at one address, but may have screwed up wallet. Not sure what my options are. Some years ago, as an experiment, I took part in a mining pool and received a number of Bitcoin payouts, all to the same address. Then I stopped mining. I didn't spend any of the balance I had accrued. 
However today for the first time in years I decided to fire up my wallet software and after sync'ing (yes, it took ages!) it appears to have zero balance. 
I can look on https://blockchain.info at the receiving address I'd originally used and see all the payments that took place originally, with the total value that I expect. But somehow it looks as though I've screwed up whatever connection there was between these and what I thought was my wallet.
I'm completely lost in the complex world of blockchain, wallets, addresses, keys etc. I thought I had all the info I needed to 'prove my ownership' of these Bitcoin payments I'd received, but clearly something's missing in what I've done.
Can anyone suggest a way that I can 're-acquire' these payments somehow? 
Apologies if the terminology I'm using isn't entirely appropriate - I seem to have forgotten whatever little I once knew about Bitcoin :(
If you do not have your private key for address you must forget about Bitcoin at all and use your national currency only. Point.

Bitcoin stands for "be your own bank". You are too dumb to understand it for serveral years.