"“He would go up very quietly to the Six Pine Trees now, peep very cautiously into the Trap, and see if there was a Heffalump there.”
The latin is literally (ignoring the cautissimē ...... esset)
he decided to creep secretly to the six pines, ..........., in order to spy out/explore.
So adrēpere,....speculātum gives the reason why he is going to the six pines and "cautissimē fundum Foveae īnspicere an Heffalumpus intus esset" says what he is going to do there. Seems natural to me....
Heffalumpus or Heffālumpus? someone more expert will have to reply on this, but my feeling is that it doesnt matter in the least.

(if you are wondering where the stress is in the word....heffa-
lum-pus?)
Persuade tibi hoc sic esse, ut scribo: quaedam tempora eripiuntur nobis, quaedam subducuntur, quaedam effluunt. Turpissima tamen est iactura, quae per neglegentiam fit. Et si volueris attendere, maxima pars vitae elabitur male agentibus, magna nihil agentibus, tota vita aliud agentibus.