yes - any insecticide which treats for aphids. Nasturtiums are much loved by blackfly (aphids) and are sometimes planted as a 'sacrifice' crop near edible plants which may get infested otherwise - the idea being that the aphids prefer the nasturtiums.

The only trouble is, if you're intending to eat the nasturtiums - assuming you're not, choose a systemic insecticide such as Bug Clear Ultra Gun or Roseclear - that will keep the plants free of infestation for up to a fortnight.

This answer assumes that what you're seeing are actually blackfly rather than real black eggs. If they are really actual eggs, a photograph would be useful.

This link might be useful to you

https://www.rhs.org.uk/advice/profile?pid=797

Unfortunately, because of the limitations of Grows on You, you will need to copy and paste it into your browser.

Thanks to Bamboo for your reply ,I will try your suggestions and hope they get rid of them.Miss sixty.

Is there caterpillar damage to any leaves; if so, it could be poo pellets - actually dark green but can appear black!

There is no caterpillar damage as I can see,I have bought a bug spray so will see if that gets rid of them. Thanks for your reply.Miss sixty.