It's not just the collapse of Thomas Cook that makes travelling between Newcastle and Arrecife (Lanzarote) a lot harder to plan than it used to be. Ryanair had already announced their intention to discontinue operating on that route from 20 January next year.
So there are just two airlines left on that route and neither of them have flights on Mondays. (So your preferred dates simply aren't possible).
TUI Airways only fly between Newcastle and Arrecife on Thursdays and they're not particularly cheap. Out on 12 March and back on 19 March costs £239 return. Out on 19 March and back on 26 March costs £245 return.
It looks like Jet2 might be your best option. They fly that route daily except Mondays and Wednesdays. Flying out on Tues 17 March and back on Tues 24 March costs £73 + £116 = £189. The following week is better, when flying out on Tues 24 March and back on Tues 31 March costs £73 + £93 = £166.
https://www.jet2.com
If you can
only travel on Mondays, you'll have to change your UK departure airport. Jet2 fly from Edinburgh to Arrecife. Flying out on 16 March and back on 23 March costs £73 + 132 = £205. Out on 23 March and back on 30 March costs £78 + £97 = £175.
If you live on the other side of Newcastle, then Leeds Bradford might be a better departure option for you. Once again it's Jet2 that operates the route to Arrecife (daily except Tues & Wed). Out on 16 March and back on 23 March costs £85 + £137 = £222. Out on 23 March and back on 30 March costs £70 + £133 = £203.