Hopefully this will get more useful soon.

• Hailesaltede, 12 places
• Halberton, 16 places
• Hallikeld, 29 places
• Ham, 3 places
• Hamestan, 27 places
• Hamfordshoe, 7 places
• Hamston, 45 places
• Happing, 18 places
• Harlow, 13 places
• Hartcliffe, 16 places
• Hartfield, 3 places
• Hartismere, 36 places
• Hartland, 14 places
• Hasler, 14 places
• Haverstoe, 22 places
• Hawkesborough, 1 place
• Headington, 31 places
• Heane, 3 places
• Helmestrei, 14 places
• Hemreswel, 1 place
• Hemyock, 15 places
• Henfield, 3 places
• Henhurst, 3 places
• Henstead, 24 places
• Hertford, 20 places
• Hessle, 21 places
• Heytesbury, 17 places
• Hezetre, 34 places
• Higham, 17 places
• Highworth, 7 places
• Hildslow, 8 places
• Hill, 18 places
• Hilton, 8 places
• Hinckford, 59 places
• Hitchin, 20 places
• Hoddington, 4 places
• Hodnet, 48 places
• Holderness [Middle Hundred], 48 places
• Holderness [North Hundred], 39 places
• Holderness [South Hundred], 39 places
• Holdshott, 9 places
• Holford, 9 places
• Holmestrow, 4 places
• Holt, 26 places
• Hoo, 2 places
• Horethorne, 18 places
• Hormer, 11 places
• Horncastle, 27 places
• Houndsborough, 11 places
• Hounslow, 2 places
• Howden, 33 places
• Humbleyard, 29 places
• Hunesberi, 11 places
• Hunthow, 20 places
• Huntspill, 3 places
• Hurstbourne, 8 places
• Hurstingstone, 18 places
• Huxloe, 9 places