Its for the child in the house. So no lap/bath use. On a table, occasionally on the sofa. The HP is tempting. But the Lenovo is slightly better specced and costs a bit less. Both seem to be similarly rated by users on the web.
My issues with HP previously have been related to warranty and heat/cooling issues.
Having integrated graphics on a Core i3, neither will suffer overheating issues, unless rested on a duvet/blanket etc whilst its busy.
I'd generally go for HP (not consumer shite like Pavillion though), Toshiba (should get one for similarish price - check Argos of all places
) or upmarket Dell (cheaper ones tend to be a bit shite). So in that "around £300" mark, I'd go HP or Tosh.
Don't settle for less than a 3rd gen Core i3 (both those in your earlier links are 4th gen), 4Gb RAM (as it'll be 64bit Windows), and if you can stretch to it, an SSD (very few laptops come with SSDs for price/spec/warranty reasons).