Oh wow, I wasn't expecting such a response.
Yes, Liz, you're on the right group of people. Son is William Goble (Cottrell) Bird, though the Cottrell is rarely seen on any kind of document. Harriet Jarrold Goble married James Bird in 1849 in Norfolk and they emigrated to the States in 1864. Harriet was in the Westhampnett area on the 1841 census, so Singleton, Westhampnett, Chi, are all pretty close together.
The John Cottrell I think is the most likely candidate was christened in Chichester on 9 Nov 1822, parents William & Ann, which would put a high likelihood that they knew each other growing up. He would only have a couple years difference in age from Harriet. This John ends up marrying Elizabeth Sinnock in 1847 in Brighton, and they live out their lives there.
The only other John Cottrell living in the Chichester area in 1841 was a Justice of the Peace and was married, and he's also quite a bit older than Harriet. Of course, this doesn't account for the remote possibility of another John Cottrell moving in the area from 1841 to 44, or the various other possibilities, however unlikely they are.
Looking for something to confirm or refute my suspicions. I think if I could find John's age, that would be one of the most helpful pieces of information. I've never dealt with Bastardy papers before, so I not sure what kind of information they actually have in them.
Oh... and yes, as far as the newspaper saying Midhurst & Chi, that's what I meant, that it says Midhurst in the description, but shows Chi.