I am working on the investigation with old documentation, to see if you can help me to recognize this town in Spain, it could be in Castilla La Mancha, although I do not know for sure.
I think it reads Enguids
Note that the last letter (an S) is a superior letter. I think it is an abbreviation of the name of the town.
So it could be Enguídanos, a town about 23km from Graja de Iniesta.
I think it reads:
natural de la Trajo de Imeita [wrong]
natural de la Graja de Iniesta término municipal de id [idéntico] provincia de Quenca
(thanks @Javi García and @Trebia Project for clarifying) but since that is not what was asked for I think the other location referenced does not start with a G because there are enough captial Gs that look different. I don't think it is an I either because if you look at "Graja de Iniesta" that would rule out an I. I rather think it is either an J or an Y. I think the third letter is rather a g than a y because there are a lot of examples of "ya" in which the y is thinner. n and u look quite similar, most probably the rounder one is the u and the edgier one is the n. What is puzzling is that in two occurrences the second letter seems to be a u and the fourth one a n while in two other occurrences this seem to be reversed. Because of that I also wouldn't rule out the possibility that both are u's.
That would mean that it is either "J/Yugnio/a" or "J/Yuguio/a".
The previous answer is not fully correct, the very first place is "Graja de Iniesta" and not "GraNja". Granja means farm in Spanish and Graja is a bird species, that's why it might look more sensible to assume a place will be named "Graja". In order to determine better the names of localities I usually double check the names here.
For the second location it is written that way when the locality if nearby the location of the register. Which is the registry source of this birth register? (based on that will be a location close to that register, or even a neighbourhood). Written is quite cryptic...