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Some thoughts, regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of an irreducible, cocommutative hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$ If the above examples are of interest for your purposes, maybe the article: Hopf algebras with one groupl-like element and these notes (especially sect. 9) will be useful.

Some thoughts, regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of an irreducible, cocommutative hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$ If the above examples are of interest for your purposes, maybe the article: Hopf algebras with one groupl-like element and these notes (especially sect. 9) will be useful.

Edit: Similar topics are discussed in most of the classical books on the subject (Sweedler's book, Abe's book, Montgomery's etc), but one of the most valuable references -imo- still remains the seminal paper of Milnor and Moore: On the structure of Hopf algebras (especially the discussion on p. 239).
(however it is important to keep in mind that in the above paper the authors define hopf algebras to be what is nowadays called "$\mathbb{Z}$-graded hopf algebras" and generally they use the terminology in a slightly different way than what is understood to be "mainstream" today).

Some thoughts, regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of a cocommutative, irreducible hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$

Some thoughts, regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of an irreducible, cocommutative hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$ If the above examples are of interest for your purposes, maybe the article: Hopf algebras with one groupl-like element and these notes (especially sect. 9) will be useful.

Regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of a cocommutative, irreducible hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$

Some thoughts, regarding question (a):

In the case of a pointed, cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: Classification of quasitriangular Hopf algebras) we have that $$ H\cong U\big(P(H)\big)\sharp kG(H) $$ where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.
(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed).

A similar result -again refering to a case with trivial grouplikes- is the situation of a cocommutative, irreducible hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism $$ H\cong U\big(P(H\big)) $$