The decays $Bs0\to J/\psi \pi +\pi -K+K-$ are studied using a data set corresponding to an integrated luminosity of 9fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, collected with the LHCb detector in proton-proton collisions at centre-of-mass energies of 7, 8 and 13TeV. The decays $Bs0\to J/\psi K\ast 0K¯\ast 0$ and $Bs0\to \chi c1(3872)K+K-$, where the $K+K-$ pair does not originate from a $\varphi$ meson, are observed for the first time. Precise measurements of the ratios of branching fractions between intermediate $\chi c1(3872)\varphi$, $J/\psi K\ast 0K¯\ast 0$, $\psi (2S)\varphi$ and $\chi c1(3872)K+K-$ states are reported. A structure, denoted as $X(4740)$, is observed in the $J/\psi \varphi$ mass spectrum and, assuming a Breit-Wigner parameterisation, its mass and width are determined to be \begin{eqnarray*} m_{X(4740)} & = & 4741 \pm 6 \pm 6,{\mathrm{MeV}}/c^2 ,, \ \Gamma_{X(4740)} & = & 53 \pm 15 \pm 11,{\mathrm{MeV}} ,, \end{eqnarray*} where the first uncertainty is statistical and the second is systematic. In addition, the most precise single measurement of the mass of the $Bs0$ meson is performed and gives a value of