My attempt:
May d. 5ten Maria Catharina, des Grenadier unter dem Hochlöb(lichen)
1ten Bataillon Heßischer Guarde, Johannes Mülhausens
fil.(ia) nata d.(en) 4ten huj.(us) h(ora) 12-1 noct.(urna) Gevatterinn des Kindes
Großmutter väterl.(icher) Seiten
Margin remark could read:
(K)ind Anna
(M)aria geb.
(M)ülhausens
the child was born between 12 and 1 at night on 4 May to Johannes Mülhausen, grenadier with the 1st Battalion of the Hessian Guards, and was baptized on 5 May