I think it's pretty common for your savings cushion to be reduced just after you've bought a house as it's such a major one-off expense. £7k sounds alright to me and I wouldn't wait to buy the house if the timing is otherwise right, but I would do a bit of a morbid/pessimistic mental exercise (get someone to help if this is going to cause any major mental/emotional wellbeing issues and maybe treat yourself to a gin or a chocolate bar afterwards!) of thinking through your personal 'worst case scenarios' to determine what your financial 'comfort level' is.
When you know this, you can build up your 'rainy day' fund again asap, then when you are there and covered for all eventualities, you know you can start to save towards other goals like home improvements or whatever.
So just as an example, my mental exercise would be running through how much money we would need to see us through disasters like a sudden death of myself or husband, severe illness or accident resulting in 6 months off work, redundancy/job loss, roof caving in, boiler breaking, car being written off or failing MOT and needing to be scrapped, all our white goods failing etc etc. The exact amounts per scenario vary depending on e.g. your work benefits and sick pay situation, whether you have any children/dependents and your lifestyle e.g. do you rely on the car for work/to live or is it a 'nice to have', and the 'answer' in some cases may not just be savings, you might want to look into critical illness insurance or boiler cover or whatever if it's more efficient. E.g. very few people could save up enough in cash to cover the total costs of raising their DC safely to adulthood so life insurance is a good idea. But most people probably could save enough to replace the boiler or the roof or to cover 3 months expenses if they lost their job or couldn't work so your 'insurance' in this scenario is to always keep enough savings...