| 1. Round | 03/08/2019 | Ayr United | - | Greenock Morton FC | 4:2 | ||||
| 2. Round | 10/08/2019 | Greenock Morton FC | - | Alloa Athletic | 4:1 | ||||
| 4. Round | 30/08/2019 | Inverness CT | - | Greenock Morton FC | 5:0 |  |
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| 5. Round | 14/09/2019 | Queen Of The South | - | Greenock Morton FC | 1:0 | |
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| 6. Round | 21/09/2019 | Greenock Morton FC | - | Dundee FC | 1:0 | ||||
| 8. Round | 05/10/2019 | Greenock Morton FC | - | Dunfermline Athletic | 1:1 | ||||
| 10. Round | 26/10/2019 | Greenock Morton FC | - | Inverness CT | 2:1 | ||||
| 11. Round | 29/10/2019 | Greenock Morton FC | - | Ayr United | 2:3 | |
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| 12. Round | 01/11/2019 | Dundee FC | - | Greenock Morton FC | 2:1 |  |
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| 13. Round | 09/11/2019 | Partick Thistle | - | Greenock Morton FC | 2:1 | ||||
| 14. Round | 16/11/2019 | Greenock Morton FC | - | Arbroath FC | 1:0 | ||||
| 16. Round | 07/12/2019 | Dunfermline Athletic | - | Greenock Morton FC | 3:1 | ||||
| 15. Round | 10/12/2019 | Greenock Morton FC | - | Dundee United | 1:2 |
England