His baptism:

Name Henry Marsh Edwards
Gender Male
Christening Date 14 Feb 1868
Christening Place OLD CHURCH,SAINT PANCRAS,LONDON,ENGLAND
Birth Date 10 Feb 1866
Father's Name Charles Hutchinson Edwards
Mother's Name Jane Louisa
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In 1901 he had a young family in Wellington, Salop where he was the incumbent:

Name Relation Condition Sex Age Birth Year Occupation  Disability  Where Born

EDWARDS, Heny M Head Married M 33 1868 Clergyman Church Of England  Northland, Hertfordshire
EDWARDS, Gertrude M Wife Married F 32 1869   Tunbridge Wells, Kent
EDWARDS, Basil H M Son M 6 1895   Tunbridge Wells, Kent
EDWARDS, James C M Son M 3 1898   Wellington, Salop
RAIKES, Eliza E Servant Single F 32 1869 Governess   Millethorpe, Westmorland
BROWN, Elizabeth Servant Single F 71 1830 Maid Domestic  Stoneham Bues, Suffolk
BATES, Annie Servant Single F 37 1864 Cook Domestic  Ellesmere, Salop
CALLCOTT, Louisa Servant Single F 19 1882 Kitchin Maid Domestic  Tregarth, Carnarvonshire

RG number: RG13 Piece: 2562 Folio: 74 Page: 52    

Registration District: Wellington Sub District: Wellington
Enumeration District: 3 Ecclesiastical Parish: All Saints

Civil Parish: Wellington Urban Municipal Borough:
Address: All Saints Vicarage, New Church Road, Wellington Urban, Town
County: Shropshire
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Later he seems to have abandoned the Church of England and joined the Old Catholics in which he was referred to as a bishop.
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In the 1911 census he was, indeed, a bishop in the Orthodox Catholic Church (probably):

First name(s) Last name Relationship Marital status  Sex  Occupation  Age  Birth year  Birth place

Henry Marsh Marsh-Edwards Head Married Male Orthodix Collislic Bishop  43  1868  London
Gertrude May Marsh-Edwards Wife Married Female -  42  1869  Tonbridge Wells
Basil Henry Marsh-Edwards Son Single Male Student  16  1895  Blackheath Kent
James Cyril Marsh-Edwards Son - Male Student 13  1898  Wellington Salop
Rode Richard Marsh-Edwards Son - Male - 9 1902  Edgbaston Warwick
Patrick John O Connell Visitor Single Male - 30  1881  London

Census details
First name(s) Henry Marsh
Last name Marsh-Edwards
Relationship Head
Marital status Married
Sex Male
Occupation Orthodix Collislic Bishop
Age 43
Birth year 1868
Birth place London
Address Dunblane Chesnut Avenue Guildford
Parish Guildford
County Surrey
Country England
Archive reference RG14
Registration district Guildford
Registration district number 33
Enumeration district 5
District number -
Sub district Guildford
Sub district number 4
Census reference RG14PN3071 RG78PN110 RD33 SD4 ED5 SN284
Piece number 3071
Folio -
Page -
Other household member first name(s) BASIL, GERTRUDE, HENRY, JAMES, PATRICK, RODE
Other household member last name MARSH-EDWARDS, O CONNELL
Record set 1911 Census for England & Wales

The name change to Marsh-Edwards must have occurred sometime beeeh the birth of their son Bede in late 1901 and this 1911 census.
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On 25 Feb 1897 this preferment was announced in the Times:

The Rev Hanry Marsh Marsh-Edwards, rector of Rollborough, vicar of Wellington-cum-Eyton, Salop--patron, Miss Georgina Hunt of Upper Norwood.
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On 14 Jan 1904, this disciplinary was noticed in the Timas:

Yesterday, the Chancellor of the diocese of Rochester (Mr J. G. Talbot) sat at Lincoln’s Omm Old Hall to resume the hearing of the charges of Immorality brought by the Bishop of Southwell under the Clergy Discipline Act 1892 against the Rev H.M. Marsh Edwards of Blackheath.  Mr Hugo Young K.C., Mr W. Appleton and Mr Kemp were counsel for the prosecution and Mr C F Gill K.C. and Mr R B Murphy represented the respondent.  Various witnesses were examined and the Court adjourned.
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On 8 Feb 1904, this notice was in The Times:

The Bishop of Rochester on Saturday pronounced the Rev Hnery Marsh Marsh-Edwards, until recently vicar of West Bridgeford, Notts, incapable of preferment.  Mr Marsh-Edwards was recently tried before a Consistory Court on a charge of misconduct with Caroline Meeson who was employed by him as a nurse girl.  The Bishop did not deal with the matter of Mr Marsh-Edwards’s deposition from holy orders.
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His death index:

First name(s) HENRY M
Last name MARSH-EDWARDS
Gender Male
Birth year 1866
Age 65
Death quarter 4
Death year 1931
District SCARBOROUGH
County Yorkshire
Volume 9D
Page 442
Country England
Record set England & Wales deaths 1837-2007
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Their marriage:

Surname First name(s) District Vol Page
Marriages Jun 1893   (>99%)
Edwards Henry Marsh Weymouth 5a 615   
Janson Gertrude Mary Weymouth 5a 615