Ah thanks. I never realised that restriction applied to piddly little local journeys like this!
It applies to all journeys made on the Network Railcard. Hence it offers effectively no discount for short journeys on weekdays. The TVM shouldn't have offered the £13 minimum fare ticket, as there are literally no circumstances under which it's worth buying for this journey.
If you regularly make short journeys at Off-Peak times, it may be worthwhile buying a Gold Card season ticket. The cheapest available one is
Hatton to Lapworth, which is £184 a year. This offers you 1/3 off as with the Network Railcard, but over a rather
larger area of validity. The time restriction is less onerous (from 09:30 vs 10:00) and importantly, there is no minimum fare. You can buy discounted tickets for up to 3 adults and 4 children travelling with you (the latter 60% off as per the Family Railcard). You can buy any other Railcard for yourself or a friend/family member for £10. It can also be applied to an Oyster card, offering the same discount on Off-Peak fares and caps as 16-25, Senior etc. Railcards.